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Encyclopedia > Logical system

Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. However the subject is grounded, the task of the logician is the same: to advance an account of valid and fallacious inference to allow one to distinguish good from bad arguments. An argument is an attempt to demonstrate the truth of an assertion called a conclusion, based on the truth of a set of assertions called premises. ...

## Nature of logic GA_googleFillSlot("encyclopedia_square");

Because of its fundamental role in philosophy, the nature of logic has been the object of intense disputation; and it is not possible to give a clear delineation of the bounds of logic in terms acceptable to all rival viewpoints. Nonetheless, the study of logic has, despite this controversy, been very coherent and technically grounded. Here we characterise logic, first by introducing the fundamental ideas about form and then by outlining some of the different schools of thought as well as giving a brief overview of its history, an account of its relationship to other sciences, and--finally--an exposition of some of logic's essential concepts.

### Informal, formal and symbolic logic

The crucial concept of form is central to discussions of the nature of logic, and it complicates exposition that 'formal' in "formal logic" is commonly used in an ambiguous manner. We shall start by giving definitions that we shall adhere to in the rest of this article:

• Informal logic is the study of natural language arguments. The study of fallacies is an especially important branch of informal logic.
• 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. We will see later that on many definitions of logic, logical inference and inference with purely formal content are the same thing. This does not render the notion of informal logic vacuous, since one may wish to investigate logic without committing to a particular formal analysis.
• Formal logic is the study of inference with purely formal content, where that content is made explicit.
• Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference.

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". 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). ... An argument is an attempt to demonstrate the truth of an assertion called a conclusion, based on the truth of a set of assertions called premises. ... A logical fallacy is an error in logical argument which is independent of the truth of the premises. ... 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. ... In mathematics, logic and computer science, a formal language is a set of finite-length words (i. ... 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. ...

While on the above analysis, formal logic is old, dating back more than two millennia, symbolic logic is comparatively new, and arises with the application of insights from mathematics to problems in logic. The passage from informal logic through formal logic to symbolic logic can be seen as a passage of increasing theoretical sophistication: of necessity, appreciating symbolic logic requires internalising certain conventions have become prevalent in the symbolic analysis of logic. Generally, the logic is captured by a formal system, comprising a formal language, which describes a set of formulas and a set of rules of derivation. The formulas will normally be intended to represent claims that we may be interested in, and likewise the rules of derivation represent inferences; such systems usually have an intended interpretation. In logic, mathematics, and computer science, a formal system is a formal grammar used for modelling purposes. ... In mathematics, logic and computer science, a formal language is a set of finite-length words (i. ...

Within this formal system, the rules of derivation and potential axioms then specify a set of theorems, which are formulas that are derivable using the rules of derivation. The most essential property of a logical formal system is soundness, which is the property that under interpretation, all of the rules of derivation are valid inferences. The theorems of a sound formal system are then truths. A minimal condition which a sound system should satisfy is consistency, meaning that no theorem contradicts another. Also important is completeness, meaning that everything true is also provable. However, when the language of logic reaches a certain degree of expressiveness (say, second-order logic), completeness becomes impossible to achieve in principle. A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ... (This article discusses the soundess notion of informal logic. ... When someone sincerely agrees with an assertion, they are claiming that it is the truth. ... 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. ... In mathematical logic, second-order logic is an extension of either propositional logic or first-order logic which contains variables in predicate positions (rather than only in term positions, as in first-order logic), and quantifiers binding them. ...

In the case of formal logical systems, the theorems are often interpretable as expressing logical truths (tautologies), and in this way can such systems be said to capture at least a part of logical truth and inference. In logic, a tautology is a statement which is true by its own definition, and is therefore fundamentally uninformative. ...

Formal logic encompasses a wide variety of logical systems. Various systems of logic we will discuss later can be captured in this framework, such as term logic, predicate logic and modal logic, and formal systems are indispensable in all branches of mathematical logic. The table of logic symbols describes various widely used notations in symbolic logic. 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 modal logic, or (less commonly) intensional logic, is a logic that deals with sentences that are qualified by modalities such as can, could, might, may, must, possibly, necessarily, eventually, etc. ... 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. ... In logic, a set of symbols is frequently used to express logical constructs. ...

### 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 Aristotle, Mozi and Aksapada Gautama, 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). Aristotle, marble copy of bronze by Lysippos. ... Mozi (c. ... Aksapada Gautama (probably c. ... 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 the judgement (German: Urteil). On this conception, the valid inferences of logic follow from the structural features of judgements or thoughts. His tomb and its pillared enclosure outside the cathedral in KÃ¶nigsberg are some of the few artifacts of German times preserved by the Soviets after they conquered East Prussia in 1945. ... Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 â€“ July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ...

A third view of logic arises from the idea that logic is more fundamental than reason, and so that logic is the science of states of affairs (German: Sachverhalt), in general. Barry Smith locates Franz Brentano as the source for this idea, an idea he claims reaches its fullest development in the work of Adolf Reinach (Smith 1989). This view of logic appears radically distinct from the first: on this conception logic has no essential connection with argument, and the study of fallacies and paradoxes no longer appears essential to the discipline. Franz Clemens Honoratus Hermann Brentano (January 16, 1838 Marienberg am Rhein (near Boppard) - March 17, 1917 ZÃ¼rich) was an influential figure in both philosophy and psychology. ... Adolf Bernhard Philipp Reinach (December 23, 1883, Mainz, Germany - November 16, 1917, Diksmuide, Belgium), German philosopher, phenomenologist (from the Munich phenomenology current) and law theorist. ...

Occasionally one encounters a fourth view as to what logic is about: it is a purely formal manipulation of symbols according to some prescribed rules. This conception can be criticized on the grounds that the manipulation of just any formal system is usually not regarded as logic. Such accounts normally omit an explanation of what it is about certain formal systems that makes them systems of logic.

### History of logic

Main articles: History of logic, and [[{{{2}}}]], and [[{{{3}}}]], and [[{{{4}}}]], and [[{{{5}}}]]

While many cultures have employed intricate systems of reasoning, logic as an explicit analysis of the methods of reasoning received sustained development originally only in three places: China in the 5th century BCE, and India and Greece between the 2nd century BCE and the 1st century BCE. The history of logic documents the development of logic as it occurs in various rival cultures and traditions in history. ... (6th century BC - 5th century BC - 4th century BC - other centuries) (2nd millennium BC - 1st millennium BC - 1st millennium AD) Events Demotic becomes the dominant script of ancient Egypt Persians invade Greece twice (Persian Wars) Battle of Marathon (490) Battle of Salamis (480) Athenian empire formed and falls Peloponnesian War... (3rd century BC - 2nd century BC - 1st century BC - other centuries) (2nd millennium BC - 1st millennium BC - 1st millennium AD) Events BC 168 Battle of Pydna -- Macedonian phalanx defeated by Romans BC 148 Rome conquers Macedonia BC 146 Rome destroys Carthage in the Third Punic War BC 146 Rome conquers... (Redirected from 1st century BCE) (2nd century BC - 1st century BC - 1st century - other centuries) The 1st century BC starts on January 1, 100 BC and ends on December 31, 1 BC. An alternative name for this century is the last century BC. (2nd millennium BC - 1st millennium BC - 1st...

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 20th century, western philosopers like Stanislaw Schayer and Klaus Glashoff have tried to explore certain aspects of Indian tradition of logic. Nyaya (pronounced as nyÎ±:yÉ™) is the name given to one of the six orthodox or astika schools of Hindu philosophy - 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. ... In 1498, the Portuguese set foot in Goa. ... 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. ...

During the medieval period, after it was shown that Aristotle's ideas were largely incompatible with faith, a greater emphasis was placed upon Aristotle's logic. During the later period of the medieval ages, logic became a main focus of philosophers, who would engage in critical logical analyses of philosophical arguments.

### Relation to other sciences

Logic is related to rationality and the structure of concepts, and so has a degree of overlap with psychology. Logic is generally understood to describe reasoning in a prescriptive manner, that is, it describes how reasoning ought to take place, however, whereas psychology is descriptive, so the overlap is not so marked. Gottlob Frege, for example, was adamant about anti-psychologism: that logic should be understood in a manner independent of the idiosyncrasies of how particular people might reason. Psychology (ancient Greek: psyche = soul or mind, logos/-ology = study of) is an academic and applied field involving the study of mind and behavior. ... Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 â€“ July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ... Anti-psychologism is a thesis about the nature of logical truth, that it does not depend upon the the contents of human ideas but exists independently. ...

### Deductive and inductive reasoning

Originally, logic consisted only of deductive reasoning which concerns what follows universally 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 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 our discussion of logic deals only with deductive logic. // Examples Valid: All men are mortal. ... This article is about induction in philosophy and logic. ... Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ... In the main, semantics (from the Greek semantikos, or significant meaning, derived from sema, sign) is the study of meaning, in some sense of that term. ... A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. ...

## 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 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. ... 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 articles: Aristotelian logic, and [[{{{2}}}]], and [[{{{3}}}]], and [[{{{4}}}]], and [[{{{5}}}]]

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 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. ... A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. ... The problem of multiple generality names a failure in Aristotelian logic to describe certain intuitively valid inferences. ...

Today, Aristotle's system is mostly seen as of historical value (though there is some current interest in extending term logics), regarded as made obsolete by the advent of the predicate calculus. First-order predicate calculus or first-order logic (FOL) permits the formulation of quantified statements such as there exists an x such that. ...

### Predicate logic

Main articles: Predicate logic, and [[{{{2}}}]], and [[{{{3}}}]], and [[{{{4}}}]], and [[{{{5}}}]]

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 parts 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 discovery 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 Friedrich Ludwig Gottlob Frege (November 8, 1848 â€“ July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ... Analytic philosophy is the dominant philosophical movement of English-speaking countries. ... First-order predicate calculus or first-order logic (FOL) permits the formulation of quantified statements such as there exists an x such that. ... Principles of Theoretical Logic is the translation into English of the seminal 1928 work of David Hilbert and Wilhelm Ackermann on the formalisation of logic, which is most well-known for introducing the now-standard formalisation of first-order logic in the Hilbert calculus. ... David Hilbert David Hilbert (January 23, 1862 â€“ February 14, 1943) was a German mathematician born in Wehlau, near KÃ¶nigsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Wilhelm Ackermann (March 29, 1896 â€“ December 24, 1962) was a German mathematician and is most famous for the Ackermann function named after him, an important example in the theory of computation. ... 1928 (MCMXXVIII) was a leap year starting on Sunday (link will take you to calendar). ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Alfred Tarski (January 14, 1901 in Warsaw â€“ October 26, 1983 in Berkeley, USA) was a Polish mathematician, and widely considered one of the four greatest logicians of all time, along with Aristotle, Gottlob Frege, and Kurt GÃ¶del. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... 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. ...

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 Shaprio. In mathematical logic, second-order logic is an extension of either propositional logic or first-order logic which contains variables in predicate positions (rather than only in term positions, as in first-order logic), and quantifiers binding them. ... W. V. Quine Willard Van Orman Quine (June 25, 1908 â€“ December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ... This article is not about George Boole, another mathematical logician. ...

### Modal logic

Main articles: Modal logic, and [[{{{2}}}]], and [[{{{3}}}]], and [[{{{4}}}]], and [[{{{5}}}]]

In language, 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. A modal logic, or (less commonly) intensional logic, is a logic that deals with sentences that are qualified by modalities such as can, could, might, may, must, possibly, necessarily, eventually, etc. ... 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, marble copy of bronze by Lysippos. ... ... 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, first put forward by Ernst Mally in 1926, is a form of modal logic used to describe and reason about obligation and permission. ... 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 Kripke in 1983 Saul Aaron Kripke (b. ... 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 — a navigational aid which lists other pages that might otherwise share the same title. ... A diagram of a graph with 6 vertices and 7 edges. ... Computational linguistics is an interdisciplinary field dealing with the statistical and logical modeling of natural language from a computational perspective. ... Wikibooks Wikiversity has more about this subject: School of Computer Science Open Directory Project: Computer Science Downloadable Science and Computer Science books Collection of Computer Science Bibliographies Belief that title science in computer science is inappropriate Categories: | ... In digital electronics, dynamic logic is sometimes used to refer to a class of design assumptions also known as clocked logic, used to distinguish this type of logic from static logic. ...

### Deduction and reasoning

Main articles: Deductive reasoning, and [[{{{2}}}]], and [[{{{3}}}]], and [[{{{4}}}]], and [[{{{5}}}]]

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. // Examples Valid: All men are mortal. ...

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. are you kiddin ? i was lookin for it for hours ...

### Mathematical logic

Main articles: Mathematical logic, and [[{{{2}}}]], and [[{{{3}}}]], and [[{{{4}}}]], and [[{{{5}}}]]

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 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. ...

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. 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 Friedrich Ludwig Gottlob Frege (November 8, 1848 â€“ July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ... Wikisource has original works written by or about: Bertrand Russell Writings available online [http://www005. ... Russells paradox (also known as Russells antinomy) is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Frege is contradictory. ... Hilberts Program was to formalize all existing theories to finite real 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. 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. Proof theory, studied as a branch of mathematical logic, represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... GÃ¶dels completeness theorem is a fundamental theorem in mathematical logic 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 models which 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. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 â€“ January 6, 1918) was a mathematician who was born in Russia and lived in Germany for most of his life. ... Note: in order to fully understand this article you may want to refer to the set theory portion of the table of mathematical symbols. ... 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. 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. Computability theory is that part of the theory of computation dealing with which problems are solvable by algorithms (equivalently, by Turing machines), with various restrictions and extensions. ... Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though professional mathematicians often treat arithmetic as a synonym for number theory. ... The Entscheidungsproblem (English: 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 Turing is often considered the father of modern computer science. ... 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 articles: Philosophical logic, and [[{{{2}}}]], and [[{{{3}}}]], and [[{{{4}}}]], and [[{{{5}}}]]

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 study of the more specifically philosophical aspects of logic. ... Philosophical logic is the study of the more specifically philosophical aspects of logic. ... 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. ... A modal logic, or (less commonly) intensional logic, is a logic that deals with sentences that are qualified by modalities such as can, could, might, may, must, possibly, necessarily, eventually, etc. ... Saul Kripke in 1983 Saul Aaron Kripke (b. ...

### Logic and computation

Main articles: Logic in computer science, and [[{{{2}}}]], and [[{{{3}}}]], and [[{{{4}}}]], and [[{{{5}}}]]

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 computer that came from this work was of fundamental importance to the designers of the computer machinery in the 1940s. This article needs to be cleaned up to conform to a higher standard of quality. ... Alan Turing is often considered the father of modern computer science. ... The Entscheidungsproblem (English: 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. ... Kurt GÃ¶del Kurt GÃ¶del [kurt gÃ¸Ëdl], (April 28, 1906 â€“ January 14, 1978) was a logician, mathematician, and philosopher of mathematics. ... In mathematical logic, GÃ¶dels incompleteness theorems are two celebrated theorems proven by Kurt GÃ¶del in 1931. ... // Events and trends World War II was a truly global conflict with many facets: immense human suffering, fierce indoctrination, and the use of new, extremely devastating weapons such as the atomic bomb. ...

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 is a programming paradigm that is claimed to be declarative (i. ... 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 logic. The ACM Computing Classification System in particular regards: Artificial intelligence (AI) is defined as intelligence exhibited by an artificial entity. ... Wikibooks Wikiversity has more about this subject: School of Computer Science Open Directory Project: Computer Science Downloadable Science and Computer Science books Collection of Computer Science Bibliographies Belief that title science in computer science is inappropriate Categories: | ... 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, formal methods refers to mathematically based techniques for the specification, development and verification of software and hardware systems (Foldoc:formalmethods). ... Hoare logic (also known as Floydâ€“Hoare logic) is a formal system developed by the British computer scientist C. A. R. Hoare, and subsequently refined by Hoare and other researchers. ... Boolean logic is a system of syllogistic logic invented by 19th-century British mathematician George Boole, which attempts to incorporate the empty set, that is, a class of non-existent entities, such as round squares, without resorting to uncertain truth values. ... A modal logic, or (less commonly) intensional logic, is a logic that deals with sentences that are qualified by modalities such as can, could, might, may, must, possibly, necessarily, eventually, etc. ... 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 is a programming paradigm that is claimed to be declarative (i. ... Automated theorem proving (currently the most important subfield of automated reasoning) is the proving of mathematical theorems by a computer program. ...

## 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 articles: classical logic, and [[{{{2}}}]], and [[{{{3}}}]], and [[{{{4}}}]], and [[{{{5}}}]]

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. ... In logic, the principle of bivalence states that for any proposition P, either P is true or P is false. ... Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ...

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. (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 1901–2000 in the sense of the Gregorian calendar (1900–1999 in the... The title given to this article is incorrect due to technical limitations. ... Ternary logic is a multi-valued logic in which there are three truth values indicating true, false and unknown. ... 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 Gentzen (November 24, 1909 – 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. However, modal logic can be used to encode non-classical logics, such as intuitionistic logic. A modal logic, or (less commonly) intensional logic, is a logic that deals with sentences that are qualified by modalities such as can, could, might, may, must, possibly, necessarily, eventually, etc. ... 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. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of a statement. ...

### Implication: strict or material?

Main articles: paradox of entailment, and [[{{{2}}}]], and [[{{{3}}}]], and [[{{{4}}}]], and [[{{{5}}}]]

It is easy to observe 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 the fact that inconsistent premises always make an argument valid; that is, inconsistent premises imply any conclusion at all. ...

The first class of paradoxes are those that involve counterfactuals, such as "If the moon is made of green cheese, then 2+2=5", puzzling because natural language does not support the principle of explosion. Eliminating these classes of paradox led to David Lewis's formulation of strict implication, and to a more radically revisionist logics such as relevance logic and dialetheism. (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. ... The name David Lewis may refer to several people: David Lewis (philosopher) (1941-2001), an American-born philosopher famous for his theory of modal realism and his love for Australia. ... 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. ... Dialetheism is a paraconsistent logic typified by its tolerance of at least some contradictions. ...

The second class of paradox are those that involve 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 modeled 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 articles: paraconsistent logics, and [[{{{2}}}]], and [[{{{3}}}]], and [[{{{4}}}]], and [[{{{5}}}]]

Closely related to questions arising from the paradoxes of implication comes the radical suggestion that logic ought to tolerate inconsistency. Again, relevance logic and dialetheism are the most important approaches here, though the concerns are different: the key issue that classical logic and some of its rivals, such as intuitionistic logic have 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 proponent of dialetheism, has argued for paraconsistency on the striking grounds that there are in fact, true contradictions (Priest 2004). A paraconsistent logic is a non-trivial logic which allows inconsistencies. ... 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. ... Dialetheism is a paraconsistent logic typified by its tolerance of at least some 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. ...

### Is logic empirical?

Main articles: Is logic empirical?, and [[{{{2}}}]], and [[{{{3}}}]], and [[{{{4}}}]], and [[{{{5}}}]]

What is the epistemological status of the laws of logic? What sort of arguments are appropriate for criticising purported principles of logic? In an influential paper entitled Is logic empirical? Hilary Putnam, building on a suggestion of W.V.O. Quine, argued that in general that 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. 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. ... These laws of classical logic are valid in propositional logic and any boolean algebra. ... Hilary Whitehall Putnam (born July 31, 1926) is a key figure in the philosophy of mind during the 20th century. ... 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. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... Philosophical realism refers to various philosophically unrelated positions, in some cases diametrically opposed ones, which are termed realism. ... The principle of distributivity states that the algebraic distributive law is valid for classical logic, where both conjunction and 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 - November 22, 1996) was an American mathematician who was born in Princeton, New Jersey, USA and died on November 22,1996 inWater Mill, New York, USA. He is the son of the mathematician George David Birkhoff. ... John von Neumann in the 1940s. ...

Another paper by the same name by Sir Michael Dummett argues that Putnam's desire for realism mandates the law of distributivity: 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. ... Metaphysics (Greek words meta = after/beyond and physics = nature) is a branch of philosophy concerned with the study of first principles and being (ontology). ... 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

• G. Birkhoff and J. von Neumann, 1936. 'The Logic of Quantum Mechanics'. Annals of Mathematics, 37:823-843.
• D. Finkelstein, 1969. 'Matter, Space and Logic'. In R. S. Cohen and M. W. Wartofsky, (eds.), Proceedings of the Boston Colloquium for the Philosophy of Science, Boston Studies in the Philosophy of Science, vol 13. ISBN 90-277-0377-9.
• D. M. Gabbay and F. Guenthner (eds.) 2001-2005. Handbook of philosophical logic (2nd ed.). 13 volumes. Dordrecht, Kluwer.
• D. Hilbert and W. Ackermann, 1928. Grundzüge der theoretischen Logik (Principles of Theoretical Logic). Springer-Verlag, ISBN 0-8218-2024-9.
• W. Hodges, 2001. Logic. An introduction to elementary logic. Penguin Books.
• T. Hofweber, 2004. Logic and Ontology. In the Stanford Encyclopedia of Philosophy.
• R. I. G. Hughes (editor), 1993. A Philosophical Companion to First-Order Logic. Hackett.
• W. Kneale and M. Kneale, 1962/1988. The Development of Logic. Oxford University Press, ISBN 0-19-824773-7.
• G. Priest, 2004. Dialetheism. In the Stanford Encyclopedia of Philosophy.
• H. Putnam, 1969. Is Logic Empirical?. Boston Studies in the Philosophy of Science, vol V.
• B. Smith, 1989. 'Logic and the Sachverhalt', The Monist, 72(1):52-69.

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. ... Principles of Theoretical Logic is the translation into English of the seminal 1928 work of David Hilbert and Wilhelm Ackermann on the formalisation of logic, which is most well-known for introducing the now-standard formalisation of first-order logic in the Hilbert calculus. ... The Stanford Encyclopedia of Philosophy (hereafter SEP) is a free online encyclopedia of philosophy run and maintained by Stanford University. ... The Stanford Encyclopedia of Philosophy (hereafter SEP) is a free online encyclopedia of philosophy run and maintained by Stanford University. ...

Results from FactBites:

 Logic - Wikipedia, the free encyclopedia (3369 words) Certain conventions have become prevalent in the symbolic analysis of logic: the logic is captured by a formal systems, comprising a formal language, which describes a set of formulas, a set of rules of derivation. Logic is extensively applied in the fields of artificial intelligence, and computer science, and these fields provide a rich source of problems in formal logic. Again, relevance logic and dialetheism are the most important approaches here, though the concerns are different: the key issue that classical logic and some of its rivals, such as intuitionistic logic have is that they respect the principle of explosion, which means that the logic collapses if it is capable of deriving a contradiction.
 John Stuart Mill's System of Logic (3880 words) Logic, however, is not the same thing with knowledge, though the field of logic is coextensive with the field of knowledge. Logic is the common judge and arbiter of all particular investigations. Logic, then, is the science of the operations of the understanding which are subservient to the estimation of evidence: both the process itself of advancing from known truths to unknown, and all other intellectual operations in so far as auxiliary to this.
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