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Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. Image File history File links Portal. ... Note: This article contains special characters. ... Look up logos in Wiktionary, the free dictionary. ... Inference is the act or process of deriving a conclusion based solely on what one already knows. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...

As a formal science, logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and through the study of arguments in natural language. The field of logic ranges from core topics such as the study of fallacies and paradoxes, to specialized analysis of reasoning using probability and to arguments involving causality. Logic is also commonly used today in argumentation theory. [1] A formal science is any one of several sciences that is predominantly concerned with abstract form, for instance, logic, mathematics, and the theoretical branches of computer science, information theory, and statistics. ... In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ... Inference is the act or process of deriving a conclusion based solely on what one already knows. ... A logical fallacy is an error in logical argument which is independent of the truth of the premises. ... Look up paradox in Wiktionary, the free dictionary. ... Probability is the likelihood that something is the case or will happen. ... It has been suggested that this article be split into multiple articles accessible from a disambiguation page. ... Argumentation theory, or argumentation, embraces the arts and sciences of civil debate, dialogue, conversation, and persuasion. ...

Traditionally, logic is studied as a branch of philosophy, one part of the classical trivium, which consisted of grammar, logic, and rhetoric. Since the mid-nineteenth century formal logic has been studied in the context of foundations of mathematics, where it was often called symbolic logic. In 1903 Alfred North Whitehead and Bertrand Russell attempted to establish logic formally as the cornerstone of mathematics with the publication of Principia Mathematica.[2] However, the system of Principia is no longer much used, having been largely supplanted by set theory. As the study of formal logic expanded, research no longer focused solely on foundational issues, and the study of several resulting areas of mathematics came to be called mathematical logic. The development of formal logic and its implementation in computing machinery is the foundation of computer science. The philosopher Socrates about to take poison hemlock as ordered by the court. ... In medieval universities, the trivium comprised the three subjects taught first: grammar, logic, and rhetoric. ... For the topic in theoretical computer science, see Formal grammar Grammar is the study of rules governing the use of language. ... Rhetoric (from Greek , rhÃªtÃ´r, orator, teacher) is generally understood to be the art or technique of persuasion through the use of spoken language; however, this definition of rhetoric has expanded greatly since rhetoric emerged as a field of study in universities. ... Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... Alfred North Whitehead, OM (February 15, 1861 Ramsgate, Kent, England â€“ December 30, 1947 Cambridge, Massachusetts, USA) was an English-born mathematician who became a philosopher. ... Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, (18 May 1872 â€“ 2 February 1970), was a British philosopher, logician, mathematician, advocate for social reform, and pacifist. ... The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ...

## Nature of logic

Form is central to logic. It complicates exposition that 'formal' in "formal logic" is commonly used in an ambiguous manner. Symbolic language is just one kind of formal logic, and is distinguished from another kind of formal logic, traditional Aristotelian syllogistic logic, which deals solely with categorical propositions. A syllogism (Greek: â€” conclusion, inference), usually the categorical syllogism, is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises) of a certain form. ... A categorical proposition is a proposition that affirms or denies a predicate of a subject. ...

• Informal logic is the study of natural language arguments. The study of fallacies is an especially important branch of informal logic. The dialogues of Plato [3] are a major example of informal logic.
• Formal logic is the study of inference with purely formal content, where that content is made explicit. (An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The first rules of formal logic that have come down to us were written by Aristotle. [4] We will see later that in many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal language captures all of the nuance of natural language.)

"Formal logic" is often used as a synonym for symbolic logic, where informal logic is then understood to mean any logical investigation that does not involve symbolic abstraction; it is this sense of 'formal' that is parallel to the received usages coming from "formal languages" or "formal theory". In the broader sense, however, formal logic is old, dating back more than two millennia, while symbolic logic is comparatively new, only about a century old. Informal logic is the study of arguments as presented in ordinary language, as contrasted with the presentations of arguments in an artificial (technical) or formal language (see formal logic). ... The term natural language is used to distinguish languages spoken and signed (by hand signals and facial expressions) by humans for general-purpose communication from constructs such as writing, computer-programming languages or the languages used in the study of formal logic, especially mathematical logic. ... In logic, an argument is a set of statements, consisting of a number of premises, a number of inferences, and a conclusion, which is said to have the following property: if the premises are true, then the conclusion must be true or highly likely to be true. ... A logical fallacy is an error in logical argument which is independent of the truth of the premises. ... PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ... Inference is the act or process of deriving a conclusion based solely on what one already knows. ... Aristotle (Greek: AristotÃ©lÄ“s) (384 BC â€“ 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ... ... Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Recursion theory, or computability theory, is a branch of mathematical logic dealing with generalizations of the notion of computable function, and with related notions such as Turing degrees and effective descriptive set theory. ... In mathematics, logic, and computer science, a formal language is a language that is defined by precise mathematical or machine processable formulas. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ...

### Consistency, soundness, and completeness

Among the valuable properties that formal systems can have are:

• Consistency, which means that none of the theorems of the system contradict one another.
• Soundness, which means that the system's rules of proof will never allow a false inference from a true premise. If a system is sound and its axioms are true then its theorems are also guaranteed to be true.
• Completeness, which means that there are no true sentences in the system that cannot, at least in principle, be proved in the system.

Not all systems achieve all three virtues. The work of Kurt Gödel has shown that no useful system of arithmetic can be both consistent and complete: see Gödel's incompleteness theorems.[5] In mathematical logic, a formal system is consistent if it does not contain a contradiction, or, more precisely, for no proposition Ï† are both Ï† and Â¬Ï† provable. ... (This article discusses the soundess notion of informal logic. ... In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ... [...]I dont believe in natural science. ... In mathematical logic, GÃ¶dels incompleteness theorems, proved by Kurt GÃ¶del in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest. ...

### Rival conceptions of logic

Logic arose (see below) from a concern with correctness of argumentation. The conception of logic as the study of argument is historically fundamental, and was how the founders of distinct traditions of logic, namely Plato and Aristotle, conceived of logic. Modern logicians usually wish to ensure that logic studies just those arguments that arise from appropriately general forms of inference; so for example the Stanford Encyclopedia of Philosophy says of logic that it "does not, however, cover good reasoning as a whole. That is the job of the theory of rationality. Rather it deals with inferences whose validity can be traced back to the formal features of the representations that are involved in that inference, be they linguistic, mental, or other representations" (Hofweber 2004). Argumentation theory, or argumentation, is the science of effective civil debate or dialogue and the effective propagation thereof, using rules of inference and logic, as applied in the real world setting. ... PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ... Aristotle (Greek: AristotÃ©lÄ“s) (384 BC â€“ 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... The Stanford Encyclopedia of Philosophy (hereafter SEP) is a free online encyclopedia of philosophy run and maintained by Stanford University. ...

By contrast Immanuel Kant introduced an alternative idea as to what logic is. He argued that logic should be conceived as the science of judgement, an idea taken up in Gottlob Frege's logical and philosophical work, where thought (German: Gedanke) is substituted for judgement (German: Urteil). On this conception, the valid inferences of logic follow from the structural features of judgements or thoughts. â€œKantâ€ redirects here. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar â€“ 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ...

### Deductive and inductive reasoning

Deductive reasoning concerns what follows necessarily from given premises. However, inductive reasoning—the process 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 of the well-understood notions of semantics. Inductive validity on the other hand requires us to define a 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 this discussion of logic deals only with deductive logic. Deductive argument follows the pattern of a general premise to a particular one, there is a very strong relationship between the premise and the conclusion of the argument. Deductive reasoning is the kind of reasoning where the conclusion is necessitated by previously known premises. ... Aristotle appears first to establish the mental behaviour of induction as a category of reasoning. ... An argument is cogent if and only if the truth of the arguments premises would render the truth of the conclusion probable (i. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. ...

## History of logic

Main article: History of logic

The first sustained work on the subject of logic which has survived was that of Aristotle. [8] The formally sophisticated treatment of modern logic descends from the Greek tradition, the latter mainly being informed from the transmission of Aristotelian logic. Aristotle (Greek: AristotÃ©lÄ“s) (384 BC â€“ 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... Aristotelian logic, also known as syllogistic logic, is the particular type of logic created by Aristotle, primarily in his works Prior Analytics and De Interpretatione. ...

The traditions outside Europe did not survive into the modern era: in China, the tradition of scholarly investigation into logic was repressed by the Qin dynasty following the legalist philosophy of Han Feizi; in the Islamic world the rise of the Asharite school suppressed original work on logic. Qin empire in 210 BC Capital Xianyang Language(s) Chinese Religion Taoism Government Monarchy History  - Unification of China 221 BC  - Death of Qin Shi Huangdi 210 BC  - Surrender to Liu Bang 206 BC The Qin Dynasty (Chinese: ; Pinyin: ; Wade-Giles: Chin Chao) (221 BC - 206 BC) was preceded... Traditional Chinese: éŸ“éžå­ Simplified Chinese: éŸ©éžå­ Pinyin: HÃ¡n FÄ“izÇ Wade-Giles: Han Fei-tzu Han Feizi (éŸ“éžå­) (d. ... The Asharite (Arabic Ø§Ù„Ø£Ø´Ø¹Ø±ÙŠØ© al-ash`aryah) is a school of early Muslim philosophy that wasinstrumental in drastically changing the direction of Islamic philosophy, separating its development radically from that of philosophy in the Christian world. ...

However in India, innovations in the scholastic school, called Nyaya, continued into the early 18th century. It did not survive long into the colonial period. In the 20th century, western philosophers like Stanislaw Schayer and Klaus Glashoff have tried to explore certain aspects of the Indian tradition of logic. According to Hermann Weyl (1929): (Sanskrit ni-ÄyÃ¡, literally recursion, used in the sense of syllogism, inference)) is the name given to one of the six orthodox or astika schools of Hindu philosophyâ€”specifically the school of logic. ... (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... It has been suggested that European colonies in India be merged into this article or section. ... The development of logic in India dates back to the analysis of inference by Aksapada Gautama, founder of the Nyaya school of Hindu philosophy, probably in the first or second centuries BCE, and so stands as one of the three original traditions of logic, alongside the Greek and Chinese traditions. ... Hermann Klaus Hugo Weyl (November 9, 1885 â€“ December 9, 1955) was a German mathematician. ...

 “ Occidental mathematics has in past centuries broken away from the Greek view and followed a course which seems to have originated in India and which has been transmitted, with additions, to us by the Arabs; in it the concept of number appears as logically prior to the concepts of geometry. ”

During the medieval period, major efforts were made to show that Aristotle's ideas were compatible with Christian faith. During the later period of the Middle Ages, logic became a main focus of philosophers, who would engage in critical logical analyses of philosophical arguments. Topics in Christianity Movements Â· Denominations Ecumenism Â· Preaching Â· Prayer Music Â· Liturgy Â· Calendar Symbols Â· Art Â· Criticism Important figures Apostle Paul Â· Church Fathers Constantine Â· Athanasius Â· Augustine Anselm Â· Aquinas Â· Palamas Â· Luther Calvin Â· Wesley Arius Â· Marcion of Sinope Pope Â· Archbishop of Canterbury Patriarch of Constantinople Christianity Portal This box:      A Christian () is a person who...

## Topics in logic

Throughout history, there has been interest in distinguishing good from bad arguments, and so logic has been studied in some more or less familiar form. Aristotelian logic has principally been concerned with teaching good argument, and is still taught with that end today, while in mathematical logic and analytical philosophy much greater emphasis is placed on logic as an object of study in its own right, and so logic is studied at a more abstract level. Aristotelian logic, also known as syllogistic logic, is the particular type of logic created by Aristotle, primarily in his works Prior Analytics and De Interpretatione. ... Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ... Analytic philosophy is the dominant philosophical movement of English-speaking countries. ...

Consideration of the different types of logic explains that logic is not studied in a vacuum. While logic often seems to provide its own motivations, the subject develops most healthily when the reason for our interest is made clear.

### Syllogistic logic

Main article: Aristotelian logic

Aristotle's work was regarded in classical times and from medieval times in Europe and the Middle East as the very picture of a fully worked out system. It was not alone: the Stoics proposed a system of propositional logic that was studied by medieval logicians; nor was the perfection of Aristotle's system undisputed; for example the problem of multiple generality was recognised in medieval times. Nonetheless, problems with syllogistic logic were not seen as being in need of revolutionary solutions. Stoicism is a school of philosophy commonly associated with such Greek philosophers as Zeno of Citium, Cleanthes, or Chrysippus and with such later Romans as Cicero, Seneca, Marcus Aurelius, and Epictetus. ... Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ... The problem of multiple generality names a failure in Aristotelian logic to describe certain intuitively valid inferences. ...

Today, some academics claim that Aristotle's system is generally seen as having little more than historical value (though there is some current interest in extending term logics), regarded as made obsolete by the advent of sentential logic and the predicate calculus. Others use Aristotle in argumentation theory to help develop and critically question argumentation schemes that are used in artificial intelligence and legal arguments. A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. ... First-order predicate calculus or first-order logic (FOL) permits the formulation of quantified statements such as there exists an x such that. ... Argumentation theory, or argumentation, embraces the arts and sciences of civil debate, dialogue, conversation, and persuasion. ... Garry Kasparov playing against Deep Blue, the first machine to win a chess game against a reigning world champion. ... This article is about law in society. ...

### Predicate logic

Main article: Predicate logic

Logic as it is studied today is a very different subject to that studied before, and the principal difference is the innovation of predicate logic. Whereas Aristotelian syllogistic logic specified the forms that the relevant part of the involved judgements took, predicate logic allows sentences to be analysed into subject and argument in several different ways, thus allowing predicate logic to solve the problem of multiple generality that had perplexed medieval logicians. With predicate logic, for the first time, logicians were able to give an account of quantifiers general enough to express all arguments occurring in natural language. ... The problem of multiple generality names a failure in Aristotelian logic to describe certain intuitively valid inferences. ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ...

The development of predicate logic is usually attributed to Gottlob Frege, who is also credited as one of the founders of analytical philosophy, but the formulation of predicate logic most often used today is the first-order logic presented in Principles of Theoretical Logic by David Hilbert and Wilhelm Ackermann in 1928. The analytical generality of the predicate logic allowed the formalisation of mathematics, and drove the investigation of set theory, allowed the development of Alfred Tarski's approach to model theory; it is no exaggeration to say that it is the foundation of modern mathematical logic. Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar â€“ 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ... Analytic philosophy is the dominant philosophical movement of English-speaking countries. ... First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ... Principles of Theoretical Logic is the title of the 1950 American translation of the 1938 second edition of David Hilberts and Wilhelm Ackermanns classic text GrundzÃ¼ge der theoretischen Logik, on elementary mathematical logic. ... David Hilbert (January 23, 1862, KÃ¶nigsberg, East Prussia â€“ February 14, 1943, GÃ¶ttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... Wilhelm Ackermann (March 29, 1896, Herscheid municipality, Germany â€“ December 24, 1962 LÃ¼denscheid, Germany ) was a German mathematician best known for the Ackermann function, an important example in the theory of computation. ... Year 1928 (MCMXXVIII) was a leap year starting on Sunday (link will display full calendar) of the Gregorian calendar. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... // Alfred Tarski (January 14, 1902, Warsaw, Russian-ruled Poland â€“ October 26, 1983, Berkeley, California) was a logician and mathematician who spent four decades as a professor of mathematics at the University of California, Berkeley. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ...

Frege's original system of predicate logic was not first-, but second-order. Second-order logic is most prominently defended (against the criticism of Willard Van Orman Quine and others) by George Boolos and Stewart Shapiro. In mathematical logic, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. ... For people named Quine, see Quine (surname). ... George Stephen Boolos (September 4, 1940, New York City - May 27, 1996) was a philosopher and a mathematical logician. ... Stewart Shapiro is Professor of Philosophy at the Ohio State University and a regular visiting professor at St. ...

### Modal logic

Main article: Modal logic

In languages, modality deals with the phenomenon that subparts of a sentence may have their semantics modified by special verbs or modal particles. For example, "We go to the games" can be modified to give "We should go to the games", and "We can go to the games"" and perhaps "We will go to the games". More abstractly, we might say that modality affects the circumstances in which we take an assertion to be satisfied. In philosophical logic, a modal logic is any logic for handling modalities: concepts like possibility, impossibility, and necessity. ... In music, modality is the subject concerning certain diatonic scales known as modes (e. ...

The logical study of modality dates back to Aristotle, who was concerned with the alethic modalities of necessity and possibility, which he observed to be dual in the sense of De Morgan duality. While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of Clarence Irving Lewis in 1918, who formulated a family of rival axiomatisations of the alethic modalities. His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to include deontic logic and epistemic logic. The seminal work of Arthur Prior applied the same formal language to treat temporal logic and paved the way for the marriage of the two subjects. Saul Kripke discovered (contemporaneously with rivals) his theory of frame semantics which revolutionised the formal technology available to modal logicians and gave a new graph-theoretic way of looking at modality that has driven many applications in computational linguistics and computer science, such as dynamic logic. Aristotle (Greek: AristotÃ©lÄ“s) (384 BC â€“ 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... ... In logic, De Morgans laws (or De Morgans theorem) are rules in formal logic relating pairs of dual logical operators in a systematic manner expressed in terms of negation. ... Clarence Irving Lewis (April 12, 1883 Stoneham, Massachusetts - February 3, 1964 Cambridge, Massachusetts) was an American academic philosopher. ... 1918 (MCMXVIII) was a common year starting on Tuesday of the Gregorian calendar (see link for calendar) or a common year starting on Wednesday of the Julian calendar. ... Deontic logic is the field of logic that is concerned with obligation, permission, and related concepts. ... Michaels the greatest boyfriend in the whole wide world, and Id love to call him in a phonebooth sometime. ... Arthur Norman Prior (1914 Masterton, New Zealand - 1969 Trondheim, Norway) was one of the foremost logicians of the twentieth century. ... In logic, the term temporal logic is used to describe any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. ... Saul Aaron Kripke (born in November, 1940, Long Beach, New York) is an American philosopher and logician now emeritus from Princeton and professor of philosophy at CUNY Graduate Center. ... Frame semantics can refer to: Kripke semantics - semantics for modal logics Frame semantics (linguistics) - linguistic theory developed by Charles Fillmore (linguist) This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share the same title. ... A pictorial representation of a graph In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. ... Computational linguistics is an interdisciplinary field dealing with the statistical and logical modeling of natural language from a computational perspective. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... Dynamic logic may mean: In modal logic: Dynamic logic is used in the context of Artificial Intelligence. ...

### Deduction and reasoning

Main article: Deductive reasoning

The motivation for the study of logic in ancient times was clear, as we have described: it is so that we may learn to distinguish good from bad arguments, and so become more effective in argument and oratory, and perhaps also, to become a better person. Deductive reasoning is the kind of reasoning where the conclusion is necessitated by previously known premises. ...

This motivation is still alive, although it no longer takes centre stage in the picture of logic; typically dialectical logic will form the heart of a course in critical thinking, a compulsory course at many universities, especially those that follow the American model. In classical philosophy, dialectic (Greek: Î´Î¹Î±Î»ÎµÎºÏ„Î¹ÎºÎ®) is an exchange of propositions (theses) and counter-propositions (antitheses) resulting in a synthesis of the opposing assertions, or at least a qualitative transformation in the direction of the dialogue. ... are you kiddin ? i was lookin for it for hours ...

### Mathematical logic

Main article: Mathematical logic

Mathematical logic really refers to two distinct areas of research: the first is the application of the techniques of formal logic to mathematics and mathematical reasoning, and the second, in the other direction, the application of mathematical techniques to the representation and analysis of formal logic. Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ...

The earliest use of mathematics and geometry in relation to logic and philosophy goes back to the ancient Greeks such as Euclid, Plato, and Aristotle. Many other ancient and medieval philosophers applied mathematical ideas and methods to their philosophical claims. Calabi-Yau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... Euclid (Greek: ), also known as Euclid of Alexandria, was a Greek mathematician of the Hellenistic period who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323 BC-283 BC). ... PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ... Aristotle (Greek: AristotÃ©lÄ“s) (384 BC â€“ 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ...

The boldest attempt to apply logic to mathematics was undoubtedly the logicism pioneered by philosopher-logicians such as Gottlob Frege and Bertrand Russell: the idea was that mathematical theories were logical tautologies, and the programme was to show this by means to a reduction of mathematics to logic.[2] The various attempts to carry this out met with a series of failures, from the crippling of Frege's project in his Grundgesetze by Russell's paradox, to the defeat of Hilbert's program by Gödel's incompleteness theorems. Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar â€“ 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ... Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, (18 May 1872 â€“ 2 February 1970), was a British philosopher, logician, mathematician, advocate for social reform, and pacifist. ... Part of the foundation of mathematics, Russells paradox (also known as Russells antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction. ... Hilberts program, formulated by German mathematician David Hilbert in the 1920s, was to formalize all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. ... In mathematical logic, GÃ¶dels incompleteness theorems are two celebrated theorems proven by Kurt GÃ¶del in 1931. ...

Both the statement of Hilbert's program and its refutation by Gödel depended upon their work establishing the second area of mathematical logic, the application of mathematics to logic in the form of proof theory.[9] Despite the negative nature of the incompleteness theorems, Gödel's completeness theorem, a result in model theory and another application of mathematics to logic, can be understood as showing how close logicism came to being true: every rigorously defined mathematical theory can be exactly captured by a first-order logical theory; Frege's proof calculus is enough to describe the whole of mathematics, though not equivalent to it. Thus we see how complementary the two areas of mathematical logic have been.[citation needed] Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... GÃ¶dels completeness theorem is an important theorem in mathematical logic which was first proved by Kurt GÃ¶del in 1929. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... Informally, we may say that a proof calculus determines a family of formal systems which specify inference rules that characterise a logical system. ...

If proof theory and model theory have been the foundation of mathematical logic, they have been but two of the four pillars of the subject. Set theory originated in the study of the infinite by Georg Cantor, and it has been the source of many of the most challenging and important issues in mathematical logic, from Cantor's theorem, through the status of the Axiom of Choice and the question of the independence of the continuum hypothesis, to the modern debate on large cardinal axioms. Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ... In Zermelo-FrÃ¤nkel set theory, Cantors theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantors theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematics, a cardinal is called a large cardinal if it belongs to a class of cardinals, the existence of which provably cannot be proved within the standard axiomatic set theory ZFC, if one assumes ZFC itself is consistent. ...

Recursion theory captures the idea of computation in logical and arithmetic terms; its most classical achievements are the undecidability of the Entscheidungsproblem by Alan Turing, and his presentation of the Church-Turing thesis.[10] Today recursion theory is mostly concerned with the more refined problem of complexity classes -- when is a problem efficiently solvable? -- and the classification of degrees of unsolvability.[11] Recursion theory, or computability theory, is a branch of mathematical logic dealing with generalizations of the notion of computable function, and with related notions such as Turing degrees and effective descriptive set theory. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... The Entscheidungsproblem (German for decision problem) is the challenge in symbolic logic to find a general algorithm which decides for given first-order statements whether they are universally valid or not. ... Alan Mathison Turing, FRS,OBE (23 June 1912 â€“ 7 June 1954) was an English mathematician, logician, and cryptographer. ... In computability theory the Church-Turing thesis, Churchs thesis, Churchs conjecture or Turings thesis, named after Alonzo Church and Alan Turing, is a hypothesis about the nature of mechanical calculation devices, such as electronic computers. ... In computational complexity theory, a complexity class is a set of problems of related complexity. ... In computability theory, the Turing degree of a subset of the natural numbers, , is the equivalence class of all subsets of equivalent to under Turing reducibility. ...

### 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., Kripke's technique of supervaluations in the semantics of logic). Philosophical logic is the application of formal logical techniques to problems that concern philosophers. ... Philosophical logic is the application of formal logical techniques to problems that concern philosophers. ... Free logic is a logic free of existential presuppositions. ... In logic, the term temporal logic is used to describe any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. ... Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ... In philosophical logic, a modal logic is any logic for handling modalities: concepts like possibility, impossibility, and necessity. ... Saul Kripke in 1983 Saul Aaron Kripke (b. ...

Logic and the philosophy of language are closely related. Philosophy of language has to do with the study of how our language engages and interacts with our thinking. Logic has an immediate impact on other areas of study. Studying logic and the relationship between logic and ordinary speech can help a person better structure their own arguments and critique the arguments of others. Many popular arguments are filled with errors because so many people are untrained in logic and unaware of how to correctly formulate an argument.

### Logic and computation

Logic cut to the heart of computer science as it emerged as a discipline: Alan Turing's work on the Entscheidungsproblem followed from Kurt Gödel's work on the incompleteness theorems, and the notion of general purpose computers that came from this work was of fundamental importance to the designers of the computer machinery in the 1940s. To meet Wikipedias quality standards, this article or section may require cleanup. ... Alan Mathison Turing, FRS,OBE (23 June 1912 â€“ 7 June 1954) was an English mathematician, logician, and cryptographer. ... The Entscheidungsproblem (German for decision problem) is the challenge in symbolic logic to find a general algorithm which decides for given first-order statements whether they are universally valid or not. ... [...]I dont believe in natural science. ... In mathematical logic, GÃ¶dels incompleteness theorems are two celebrated theorems proven by Kurt GÃ¶del in 1931. ... This article or section does not cite any references or sources. ...

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. In logic programming, a program consists of a set of axioms and rules. Logic programming systems such as Prolog compute the consequences of the axioms and rules in order to answer a query. Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. ... Logic programming (which might better be called logical programming by analogy with mathematical programming and linear programming) is, in its broadest sense, the use of mathematical logic for computer programming. ... Prolog is a logic programming language. ...

Today, logic is extensively applied in the fields of artificial intelligence, and computer science, and these fields provide a rich source of problems in formal and informal logic. Argumentation theory is one good example of how logic is being applied to artificial intelligence. The ACM Computing Classification System in particular regards: Garry Kasparov playing against Deep Blue, the first machine to win a chess game against a reigning world champion. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... Argumentation theory, or argumentation, embraces the arts and sciences of civil debate, dialogue, conversation, and persuasion. ... The ACM Computing Classification System is a subject classification system for computer science devised by the Association for Computing Machinery. ...

Furthermore, computers can be used as tools for logicians. For example, 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 theoretical computer science formal semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages and models of computation. ... In computer science and software engineering, formal methods are mathematically-based techniques for the specification, development and verification of software and hardware systems. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... Boolean logic is a complete system for logical operations. ... In philosophical logic, a modal logic is any logic for handling modalities: concepts like possibility, impossibility, and necessity. ... Default logic is a non-monotonic logic proposed by Ray Reiter to formalize the way humans reason using default assumptions. ... Knowledge representation formalisms and methods is the name of section I.2. ... In logic, and in particular in propositional calculus, a Horn clause is a proposition of the general type (p and q and . ... Logic programming (which might better be called logical programming by analogy with mathematical programming and linear programming) is, in its broadest sense, the use of mathematical logic for computer programming. ... Description logics (DL) are a family of knowledge representation languages which can be used to represent the terminological knowledge of an application domain in a structured and formally well-understood way. ... Automated theorem proving (ATP) or automated deduction, currently the most well-developed subfield of automated reasoning (AR), is the proving of mathematical theorems by a computer program. ...

### Argumentation theory

Argumentation theory is the study and research of informal logic, fallacies, and critical questions as they relate to every day and practical situations. Specific types of dialogue can be analyzed and questioned to reveal premises, conclusions, and fallacies. Argumentation theory is now applied in artificial intelligence and law. Argumentation theory, or argumentation, is the science of effective civil debate or dialogue and the effective propagation thereof, using rules of inference and logic, as applied in the real world setting. ... Garry Kasparov playing against Deep Blue, the first machine to win a chess game against a reigning world champion. ... Lady Justice or Justitia is a personification of the moral force that underlies the legal system (particularly in Western art). ...

## Controversies in logic

Just as we have seen there is disagreement over what logic is about, so there is disagreement about what logical truths there are.

### Bivalence and the law of the excluded middle

Main article: Classical logic

The logics discussed above are all "bivalent" or "two-valued"; that is, they are most naturally understood as dividing propositions into the true and the false propositions. Systems which reject bivalence are known as non-classical logics. Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ... Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ...

In 1910 Nicolai A. Vasiliev rejected the law of excluded middle and the law of contradiction and proposed the law of excluded fourth and logic tolerant to contradiction. In the early 20th century Jan Łukasiewicz investigated the extension of the traditional true/false values to include a third value, "possible", so inventing ternary logic, the first multi-valued logic. Vasiliev, Nicolai Alexandrovich (Vasilev, Vassilieff, Wassilieff), (1880 â€“ 1940) was a Russian logician, philosopher, psychologist, poet, the forerunner of paraconsistent and multi-valued logics. ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901&#8211;2000 in the sense of the Gregorian calendar (1900&#8211;1999... // Jan Åukasiewicz (21 December 1878 - 13 February 1956) was a Polish mathematician born in Lemberg, Galicia, Austria-Hungary (now Lviv, Ukraine). ... A ternary, three-valued or trivalent logic is a term to describe any of several multi-valued logic systems in which there are three truth values indicating true, false and some third value. ... Multi-valued logics are logical calculi in which there are more than two possible truth values. ...

Intuitionistic logic was proposed by L.E.J. Brouwer as the correct logic for reasoning about mathematics, based upon his rejection of the law of the excluded middle as part of his intuitionism. Brouwer rejected formalisation in mathematics, but his student Arend Heyting studied intuitionistic logic formally, as did Gerhard Gentzen. Intuitionistic logic has come to be of great interest to computer scientists, as it is a constructive logic, and is hence a logic of what computers can do. Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, and measure theory and complex analysis. ... The law of excluded middle (tertium non datur in Latin) states that for any proposition P, it is true that (P or ~P). ... In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ... Arend Heyting (May 9, 1898 â€“ July 9, 1980) was a Dutch mathematician and logician. ... Gerhard Karl Erich Gentzen (November 24, 1909 â€“ August 4, 1945) was a German mathematician and logician. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ...

Modal logic is not truth conditional, and so it has often been proposed as a non-classical logic. However, modal logic is normally formalised with the principle of the excluded middle, and its relational semantics is bivalent, so this inclusion is disputable. On the other hand, modal logic can be used to encode non-classical logics, such as intuitionistic logic. In philosophical logic, a modal logic is any logic for handling modalities: concepts like possibility, impossibility, and necessity. ... Kripke semantics (also known as possible world semantics, relational semantics, or frame semantics) is a formal semantics for modal logic systems, created in late 1950s and early 1960s by Saul Kripke. ...

Logics such as fuzzy logic have since been devised with an infinite number of "degrees of truth", 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. Fuzzy logic is derived from fuzzy set theory dealing with reasoning that is approximate rather than precisely deduced from classical predicate logic. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Bayesian probability is an interpretation of probability suggested by Bayesian theory, which holds that the concept of probability can be defined as the degree to which a person believes a proposition. ...

### Implication: strict or material?

Main article: Paradox of entailment

It is obvious that the notion of implication formalised in classical logic does not comfortably translate into natural language by means of "if... then...", due to a number of problems called the paradoxes of material implication. The paradox of entailment is an apparent paradox derived from the observation that, in classical logic, inconsistent premises always make an argument valid; that is, inconsistent premises imply any conclusion at all. ...

The first class of paradoxes involves counterfactuals, such as "If the moon is made of green cheese, then 2+2=5", which are puzzling because natural language does not support the principle of explosion. Eliminating this class of paradoxes was the reason for C. I. Lewis's formulation of strict implication, which eventually led to more radically revisionist logics such as relevance logic. (A âˆ§ Â¬A)â†’ B Ex falso quodlibet, also known as ex contradictione (sequitur) quodlibet or the principle of explosion is the rule of classical logic that states that anything follows from a contradiction. ... Clarence Irving Lewis (April 12, 1883 _ February 3, 1964) was a pragmatist philosopher. ... In logic, a strict conditional is a material conditional that is acted upon by the necessity operator from modal logic. ... Relevance logic, also called relevant logic, is any of a family of non-classical substructural logics that impose certain restrictions on implication. ...

The second class of paradoxes involves redundant premises, falsely suggesting that we know the succedent because of the antecedent: thus "if that man gets elected, granny will die" is materially true if granny happens to be in the last stages of a terminal illness, regardless of the man's election prospects. Such sentences violate the Gricean maxim of relevance, and can be modelled by logics that reject the principle of monotonicity of entailment, such as relevance logic. Paul Grice, the philosopher, proposed four conversational maxims that arise from the pragmatics of natural language. ... Monotonicity of entailment - Wikipedia /**/ @import /w/skins-1. ...

### Tolerating the impossible

Main article: Paraconsistent logic

Closely related to questions arising from the paradoxes of implication comes the radical suggestion that logic ought to tolerate inconsistency. Relevance logic and paraconsistent logic are the most important approaches here, though the concerns are different: a key consequence of classical logic and some of its rivals, such as intuitionistic logic, is that they respect the principle of explosion, which means that the logic collapses if it is capable of deriving a contradiction. Graham Priest, the main proponent of dialetheism, has argued for paraconsistency on the grounds that there are in fact, true contradictions.[12] A paraconsistent logic is a logical system that attempts to deal nontrivially with contradictions. ... Consistency has three technical meanings: In mathematics and logic, as well as in theoretical physics, it refers to the proposition that a formal theory or a physical theory contains no contradictions. ... Relevance logic, also called relevant logic, is any of a family of non-classical substructural logics that impose certain restrictions on implication. ... A paraconsistent logic is a logical system that attempts to deal nontrivially with contradictions. ... Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... (A âˆ§ Â¬A)â†’ B Ex falso quodlibet, also known as ex contradictione (sequitur) quodlibet or the principle of explosion is the rule of classical logic that states that anything follows from a contradiction. ... Graham (Grammy) Priest (born 1948) is Boyce Gibson Professor of Philosophy at the University of Melbourne and a regular visitor at St. ... Dialetheism is a paraconsistent logic typified by its tolerance of at least some contradictions. ...

### Is logic empirical?

Main article: Is logic empirical?

What is the epistemological status of the laws of logic? What sort of argument is appropriate for criticising purported principles of logic? In an influential paper entitled "Is logic empirical?"[13] Hilary Putnam, building on a suggestion of W.V. Quine, argued that in general the facts of propositional logic have a similar epistemological status as facts about the physical universe, for example as the laws of mechanics or of general relativity, and in particular that what physicists have learned about quantum mechanics provides a compelling case for abandoning certain familiar principles of classical logic: if we want to be realists about the physical phenomena described by quantum theory, then we should abandon the principle of distributivity, substituting for classical logic the quantum logic proposed by Garrett Birkhoff and John von Neumann.[14] Is logic empirical? is the title of two articles that discuss the radical concept, that the empirical facts about quantum phenomena may provide grounds for revising classical logic. ... It has been suggested that Meta-epistemology be merged into this article or section. ... These laws of classical logic are valid in propositional logic and any boolean algebra. ... Hilary Whitehall Putnam (born July 31, 1926) is an American philosopher who has been a central figure in Western philosophy since the 1960s, especially in philosophy of mind, philosophy of language, and philosophy of science. ... W. V. Quine Willard Van Orman Quine (June 25, 1908 - December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ... Mechanics (Greek ) is the branch of physics concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effect of the bodies on their environment. ... An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... Contemporary philosophical realism, also referred to as metaphysical realism, is the belief in a reality that is completely ontologically independent of our conceptual schemes, linguistic practices, beliefs, etc. ... The principle of distributivity states that the algebraic distributive law is valid for classical logic, where both logical conjunction and logical disjunction are distributive over each other. ... In mathematical physics and quantum mechanics, quantum logic can be regarded as a kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables. ... Garrett Birkhoff (January 19, 1911, Princeton, New Jersey, USA - November 22, 1996, Water Mill, New York, USA) was an American mathematician. ... John von Neumann (Hungarian Margittai Neumann JÃ¡nos Lajos) (born December 28, 1903 in Budapest, Austria-Hungary; died February 8, 1957 in Washington D.C., United States) was a Hungarian-born American mathematician who made contributions to quantum physics, functional analysis, set theory, topology, economics, computer science, numerical analysis, hydrodynamics...

Another paper by the same name by Sir Michael Dummett argues that Putnam's desire for realism mandates the law of distributivity.[15] Distributivity of logic is essential for the realist's understanding of how propositions are true of the world in just the same way as he has argued the principle of bivalence is. In this way, the question, "Is logic empirical?" can be seen to lead naturally into the fundamental controversy in metaphysics on realism versus anti-realism. Sir Michael A. E. Dummett (1925 - ) is a leading British philosopher, who has both written on the history of analytic philosophy, and made original contributions to the subject, particularly in the areas of philosophy of mathematics, philosophy of logic, philosophy of language and metaphysics. ... Plato (Left) and Aristotle (right), by Raphael (Stanza della Segnatura, Rome) Metaphysics is the branch of philosophy concerned with explaining the ultimate nature of reality, being, and the world. ... In philosophy, the term anti-realism is used to describe any position involving either the denial of the objective reality of entities of a certain type or the insistence that we should be agnostic about their real existence. ...

## References

• Brookshear, J. Glenn (1989), Theory of computation : formal languages, automata, and complexity, Benjamin/Cummings Pub. Co., Redwood City, Calif. ISBN 0805301437
• Cohen, R.S, and Wartofsky, M.W. (1974), Logical and Epistemological Studies in Contemporary Physics, Boston Studies in the Philosophy of Science, D. Reidel Publishing Company, Dordrecht, Netherlands. ISBN 90-277-0377-9.
• Finkelstein, D. (1969), "Matter, Space, and Logic", in R.S. Cohen and M.W. Wartofsky (eds. 1974).
• Gabbay, D.M., and Guenthner, F. (eds., 2001-2005), Handbook of Philosophical Logic, 13 vols., 2nd edition, Kluwer Publishers, Dordrecht.
• Vincent F. Hendricks, Thought 2 Talk: A Crash Course in Reflection and Expression, New York: Automatic Press / VIP, 2005, ISBN 87-991013-7-8.
• Hilbert, D., and Ackermann, W. (1928), Grundzüge der theoretischen Logik (Principles of Theoretical Logic), Springer-Verlag. OCLC 2085765
• Hodges, W. (2001), Logic. An introduction to Elementary Logic, Penguin Books.
• Hofweber, T. (2004), "Logic and Ontology", Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), Eprint.
• Hughes, R.I.G. (ed., 1993), A Philosophical Companion to First-Order Logic, Hackett Publishing.
• Kneale, William, and Kneale, Martha, (1962), The Development of Logic, Oxford University Press, London, UK.
• Mendelson, Elliott (1964), Introduction to Mathematical Logic, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, Calif. OCLC 13580200
• Smith, B. (1989), "Logic and the Sachverhalt", The Monist 72(1), 52–69.
• Whitehead, Alfred North and Bertrand Russell (1910), Principia Mathematica, The University Press, Cambridge, England. OCLC 1041146

Vincent F. Hendricks is a philosopher and logician. ... David Hilbert (January 23, 1862, KÃ¶nigsberg, East Prussia â€“ February 14, 1943, GÃ¶ttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... Wilhelm Ackermann (March 29, 1896, Herscheid municipality, Germany â€“ December 24, 1962 LÃ¼denscheid, Germany ) was a German mathematician best known for the Ackermann function, an important example in the theory of computation. ... Principles of Theoretical Logic is the title of the 1950 American translation of the 1938 second edition of David Hilberts and Wilhelm Ackermanns classic text GrundzÃ¼ge der theoretischen Logik, on elementary mathematical logic. ... The Stanford Encyclopedia of Philosophy (hereafter SEP) is a free online encyclopedia of philosophy run and maintained by Stanford University. ... Edward N. Zalta is a Senior Research Scholar at the Center for the Study of Language and Information. ... Several notable people are named Barry Smith: Barry Smith, an ontologist at the University at Buffalo, The State University of New York Barry Windsor-Smith, a comics artist Barry Thomas Smith, a comics artist Barry Smith (musician) Barry Smith, preacher from New Zealand Barry Smith (AKA Barry Seven), former member... Alfred North Whitehead, OM (February 15, 1861 Ramsgate, Kent, England â€“ December 30, 1947 Cambridge, Massachusetts, USA) was an English-born mathematician who became a philosopher. ... Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, (18 May 1872 â€“ 2 February 1970), was a British philosopher, logician, mathematician, advocate for social reform, and pacifist. ... The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ...

## Notes

1. ^ J. Robert Cox and Charles Arthur Willard, eds. Advances in Argumentation Theory and Research, Southern Illinois University Press, 1983 ISBN 0809310503, ISBN-13 978-0809310500
2. ^ a b c Alfred North Whitehead and Bertrand Russell, Principia Mathematical to *56, Cambridge University Press, 1967, ISBN 0-521-62606-4
3. ^ Plato, The Portable Plato, edited by Scott Buchanan, Penguin, 1976, ISBN 0-14-015040-4
4. ^ Aristotle, The Basic Works, Richard Mckeon, editor, Modern Library, 2001, ISBN 0-375-75799-6, see especially, Posterior Analytics.
5. ^ a b For a more modern treatment, see A. G. Hamilton, Logic for Mathematicians, Cambridge, 1980, ISBN 0-521-29291-3
6. ^ S. Kak (2004). The Architecture of Knowledge. CSC, Delhi.
7. ^ McGreal 1995, p. 33
8. ^ Morris Kline, "Mathematical Thought From Ancient to Modern Times, Oxford University Press, 1972, ISBN 0-19-506135-7, p.53 "A major achievement of Aristotle was the founding of the science of logic."
9. ^ Mendelson, "Formal Number Theory: Gödel's Incompleteness Theorem"
10. ^ Brookshear, "Computability: Foundations of Recursive Function Theory"
11. ^ Brookshear, "Complexity"
12. ^ Priest, Graham (2004), "Dialetheism", Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), http://plato.stanford.edu/entries/dialetheism.
13. ^ Putnam, H. (1969), "Is Logic Empirical?", Boston Studies in the Philosophy of Science. 5.
14. ^ Birkhoff, G., and von Neumann, J. (1936), "The Logic of Quantum Mechanics", Annals of Mathematics 37, 823–843.
15. ^ Dummett, M. (1978), "Is Logic Empirical?", Truth and Other Enigmas. ISBN 0-674-91076-1

Posterior Analytics (or Analytica Posteriora) is a text by Aristotle. ... Subhash Kak (à¤¸à¥à¤­à¤¾à¤· à¤•à¤¾à¤•) (born March 26, 1947, Srinagar, Kashmir) is Delaune Distinguished Professor of Electrical Engineering and Professor in the Asian Studies and Cognitive Science Programs at Louisiana State University, Baton Rouge. ... Graham (Grammy) Priest (born 1948) is Boyce Gibson Professor of Philosophy at the University of Melbourne and a regular visitor at St. ... The Annals of Mathematics (ISSN 0003-486X), often just called Annals, is a bimonthly mathematics research journal published by Princeton University and the Institute for Advanced Study. ...

Charles Lutwidge Dodgson (Lewis Carroll) â€“ believed to be a self-portrait Charles Lutwidge Dodgson (IPA: ) (January 27, 1832 â€“ January 14, 1898), better known by the pen name Lewis Carroll, was an English author, mathematician, logician, Anglican clergyman and photographer. ... Michael Scriven is an academic who has made significant contributions in the fields of philosophy, psychology, and perhaps most notably, educational evaluation. ... Susan Haack (born 1945) is an English professor of philosophy and law at the University of Miami in Coral Gables, Florida, in the United States. ... Nicholas Rescher (born July 15, 1928 in Hagen, Germany) is an American philosopher, affiliated for many years with the University of Pittsburgh, where he is currently University Professor of Philosophy and Chairman of the Center for the Philosophy of Science. ...

Garry Kasparov playing against Deep Blue, the first machine to win a chess game against a reigning world champion. ... Deductive reasoning is the kind of reasoning where the conclusion is necessitated by previously known premises. ... Digital circuits are electric circuits based on a number of discrete voltage levels. ... The development of logic in India dates back to the analysis of inference by Aksapada Gautama, founder of the Nyaya school of Hindu philosophy, probably in the first or second centuries BCE, and so stands as one of the three original traditions of logic, alongside the Greek and Chinese traditions. ... Aristotle appears first to establish the mental behaviour of induction as a category of reasoning. ... Logical consequence is the relation that holds between a set of sentences and a sentence when the latter follows from the former. ... A logic puzzle is a puzzle deriving from the mathematics field of deduction. ... Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Mathematics is the search for fundamental truths in pattern, quantity, and change. ... These list of mathematics articles pages collect pointers to all articles related to mathematics. ... The philosopher Socrates about to take poison hemlock as ordered by the court. ... Philosophy is a broad field of knowledge in which the definition of knowledge itself is one of the subjects investigated. ... ... The aim of a probabilistic logic (or probability logic) is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure. ... Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ... For other uses, see Reason (disambiguation). ... In logic, a set of symbols is frequently used to express logical constructs. ... Traditional logic, also known as term logic, is a loose term for the logical tradition that originated with Aristotle and survived broadly unchanged until the advent of modern predicate logic in the late nineteenth century. ... A common dictionary definition of truth is agreement with fact or reality.[1] There is no single definition of truth about which the majority of philosophers agree. ... Sojourner Truth A truth theory or a theory of truth is a conceptual framework that underlies a particular conception of truth, such as those used in art, ethics, logic, mathematics, philosophy, the sciences, or any discussion that either mentions or makes use of a notion of truth. ...

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 Apple - Logic Express (1939 words) Logic Express 8 delivers the power, precision, and professional toolset of Logic Pro 8—including a redesigned interface that allows musicians to write, record, edit, and mix with unparalleled speed and ease. Logic Express 8 features 36 incredible-sounding instrument plug-ins from Logic Studio—including Ultrabeat, EXS24 Sampler, the ES synth series, and more than a dozen GarageBand instruments. Logic Express can be used with any audio interface that works with Mac OS X, so you never have to worry about restrictive proprietary hardware requirements.
 Logic - Wikipedia, the free encyclopedia (4685 words) The ambiguity is that "formal logic" is very often used with the alternate meaning of symbolic logic as we have defined it, with informal logic meaning any logical investigation that does not involve symbolic abstraction; it is this sense of 'formal' that is parallel to the received usages coming from "formal languages" or "formal theory". Logic cut to the heart of computer science as it emerged as a discipline: Alan Turing's work on the Entscheidungsproblem followed from Kurt Gödel's work on the incompleteness theorems, and the notion of general purpose computers that came from this work was of fundamental importance to the designers of the computer machinery in the 1940s. Relevance logic and paraconsistent logic are the most important approaches here, though the concerns are different: a key consequence of classical logic and some of its rivals, such as intuitionistic logic, is that they respect the principle of explosion, which means that the logic collapses if it is capable of deriving a contradiction.
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