Logic is the study of the principles of valid inference and demonstration. Inference is the act or process of deriving a Conclusion based solely on what one already knows In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true The word derives from Greek λογική (logike), fem. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly of λογικός (logikos), "possessed of reason, intellectual, dialectical, argumentative", from λόγος logos, "word, thought, idea, argument, account, reason, or principle". grc-Latn Logos (ˈloʊːgɒs ( Greek, logos) is an important term in Philosophy, Analytical psychology, Rhetoric and Religion [1][2]

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. A formal science is a theoretical study that is concerned with theoretical Formal systems, for instance Logic, Mathematics, Systems theory and In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence In Logic, an argument is a Set of one or more Declarative sentences (or "propositions") known as the Premises along In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist Inference is the act or process of deriving a Conclusion based solely on what one already knows The field of logic ranges from core topics such as the study of validity, fallacies and paradoxes, to specialized analysis of reasoning using probability and to arguments involving causality. The term validity (also called logical truth, analytic truth, or necessary truth) as it occurs in Logic refers generally to a property of A fallacy is a component of an Argument which being demonstrably flawed in its Logic or form renders the argument invalid in whole A paradox is a true statement or group of statements that leads to a Contradiction or a situation which defies intuition; or inversely Probability is the likelihood or chance that something is the case or will happen Causality (but not causation) denotes a necessary relationship between one event (called cause and another event (called effect) which is the direct consequence Logic is also commonly used today in argumentation theory. Argumentation theory, or argumentation, embraces the arts and sciences of civil debate Dialogue, conversation and persuasion studying rules of Inference [3]

Traditionally, logic was considered a branch of philosophy, a part of the classical trivium of grammar, logic, and rhetoric. Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language In medieval universities, the trivium comprised the three subjects taught first Grammar, Logic, and Rhetoric. Grammar is the field of Linguistics that covers the Rules governing the use of any given natural language. Rhetoric has had many definitions no simple definition can do it justice Since the mid-nineteenth century formal logic has been studied in the context of foundations of mathematics, where it was often called symbolic logic. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. Foundations of mathematics is a term sometimes used for certain fields of Mathematics, such as Mathematical logic, Axiomatic set theory, Proof theory Symbolic logic is the area of Mathematics which studies the purely formal properties of strings of symbols In 1879 Frege published Begriffsschrift : A formula language or pure thought modelled on that of arithemetic which inaugurated modern logic with the invention of quantifier notation. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin  &ndash 26 July 1925 Begriffsschrift is the title of a short book on Logic by Gottlob Frege, published in 1879, and is also the name of the Formal system Quantification has two distinct meanings In Mathematics and Empirical science, it refers to human acts known as Counting and Measuring In 1903 Alfred North Whitehead and Bertrand Russell attempted to establish logic formally as the cornerstone of mathematics with the publication of Principia Mathematica. Alfred North Whitehead, OM ( February 15 1861, Ramsgate, Kent, England &ndash December 30 1947, Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian The Principia Mathematica is a 3-volume work on the Foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell [4] However, except for the elementary part, the system of Principia is no longer much used, having been largely superseded by set theory. At the same time the developments in the field of Logic since Frege, Russell and Wittgenstein had a profound influence on both the practice of philosophy and the ideas concerning the nature of philosophical problems especially in the English speaking world (see Analytic philosophy). Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin  &ndash 26 July 1925 Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian Analytic philosophy (sometimes analytical philosophy) is a generic term for a style of Philosophy that came to dominate English-speaking countries in the 20th century 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. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. The development of formal logic and its implementation in computing machinery is fundamental to computer science. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Logic is now widely taught by university philosophy departments, more often than not as a compulsory discipline for their students, especially in the English speaking world.

## Nature of logic

Form is central to logic. It complicates exposition that 'formal' in "formal logic" is commonly used in an ambiguous manner. Symbolic logic 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, or logical appeal, (συλλογισμός &mdash "conclusion" "inference" (usually the categorical syllogism) is a kind of A CATEGORICAL PROPOSITION is what gives a direct assertion of agreement or disagreement between the subject term and predicate term

• Informal logic is the study of natural language arguments. Informal logic (or occasionally non-formal logic) is the study of arguments as presented in ordinary language as contrasted with the presentations of arguments in In the Philosophy of language, a natural language (or ordinary language) is a Language that is spoken or written in phonemic-alphabetic or phonemically-related In Logic, an argument is a Set of one or more Declarative sentences (or "propositions") known as the Premises along The study of fallacies is an especially important branch of informal logic. A fallacy is a component of an Argument which being demonstrably flawed in its Logic or form renders the argument invalid in whole The dialogues of Plato[5] are a good example of informal logic. Biography Early life Birth and family Plato was born in Athens Greece
• Formal logic is the study of inference with purely formal content, where that content is made explicit. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. Inference is the act or process of deriving a Conclusion based solely on what one already knows (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 works of Aristotle contain the earliest known formal study of logic, which were incorporated in the late nineteenth century into modern formal logic. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. [6] 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 logic captures all of the nuance of natural language. )
• Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is the area of Mathematics which studies the purely formal properties of strings of symbols [4][7] Symbolic logic is often divided into two branches, propositional logic and predicate logic. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" In Mathematical logic, predicate logic is the generic term for symbolic Formal systems like First-order logic, Second-order logic, Many-sorted
• Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of model theory, proof theory, set theory, and recursion theory. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models Proof theory is a branch of Mathematical logic that represents proofs as formal Mathematical objects facilitating their analysis by mathematical techniques Recursion theory, also called computability theory, is a branch of Mathematical logic that originated in the 1930s with the study of Computable functions

"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". A formal language is a set of words, ie finite strings of letters, or symbols. 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 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.

### Consistency, soundness, and completeness

Among the valuable properties that logical systems can have are:

• Consistency, which means that none of the theorems of the system contradict one another. In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist
• Soundness, which means that the system's rules of proof will never allow a false inference from a true premise. In Mathematical logic, a Logical system has the soundness property If and only if its Inference rules prove only formulas that are 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. In general an object is complete if nothing needs to be added to it

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. Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most [7]

### Rival conceptions of logic

Logic arose (see below) from a concern with correctness of argumentation. Argumentation theory, or argumentation, embraces the arts and sciences of civil debate Dialogue, conversation and persuasion studying rules of Inference 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. The Stanford Encyclopedia of Philosophy (SEP is a freely-accessible Online encyclopedia of Philosophy maintained by Stanford University. 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).

By contrast, Immanuel Kant argued that logic should be conceived as the science of judgment, an idea taken up in Gottlob Frege's logical and philosophical work, where thought (German: Gedanke) is substituted for judgement (German: Urteil). Immanuel Kant (ɪmanuəl kant 22 April 1724 12 February 1804 was an 18th-century German Philosopher from the Prussian city of Königsberg Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin  &ndash 26 July 1925 On this conception, the valid inferences of logic follow from the structural features of judgements or thoughts.

### Deductive and inductive reasoning

Deductive reasoning concerns what follows necessarily from given premises. Deductive reasoning is Reasoning which uses deductive Arguments to move from given statements ( Premises to Conclusions which must be true if the However, inductive reasoning—the process of deriving a reliable generalization from observations—has sometimes been included in the study of logic. Induction or inductive reasoning, sometimes called inductive logic, is the process of Reasoning in which the premises of an argument are believed Correspondingly, we must distinguish between deductive validity and inductive validity (called "cogency"). An argument is cogent if and only if the truth of the argument's premises would render the truth of the conclusion probable (i 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. Semantics is the study of meaning in communication The word derives from Greek σημαντικός ( semantikos) "significant" from 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. Note The term model has a different meaning in Model theory, a branch of Mathematical logic. 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.

## History of logic

Main article: History of logic

Several ancient civilizations have employed intricate systems of reasoning and asked questions about logic or propounded logical paradoxes. The history of logic traces the development of the science of valid inference ( Logic) In India, the Nasadiya Sukta of the Rigveda (RV 10. The development of Indian logic can be said to date back to the anviksiki of Medhatithi Gautama (c The Nasadiya Sukta (after the incipit ná ásat "not the non-existent" is the 129th hymn of the 10th Mandala of the Rigveda. The Rigveda ( Sanskrit sa ऋग्वेद ṛgveda, a compound of ṛc "praise verse" and veda "knowledge" The tenth Mandala of the Rigveda has 191 hymns Together with Mandala 1, it forms the latest part of the Rigveda containing much mythological material 129) contains ontological speculation in terms of various logical divisions that were later recast formally as the four circles of catuskoti: "A", "not A", "A and not A", and "not A and not not A". In Philosophy, ontology (from the Greek, genitive: of being (part The tetralemma ( catuskoti) is a figure that features prominently in Indian traditional logic [8] The Chinese philosopher Gongsun Long (ca. Gongsun Long ( ca 325–250 BC was a member of the Logicians school of ancient Chinese philosophy. 325250 BC) proposed the paradox "One and one cannot become two, since neither becomes two. Events By place Macedonian Empire Alexander the Great leaves India and nominates his officer Peithon son of Agenor Events By place Egypt Ptolemy II encourages the Jewish residents of Alexandria to have their Bible translated "[9] In China, the tradition of scholarly investigation into logic, however, was repressed by the Qin dynasty following the legalist philosophy of Han Feizi. Not to be confused with the Qing Dynasty, the last dynasty of China Han Fei (also Han Feizi) ( (ca 280&ndash233 BC was a Philosopher who along with Li Si, developed Xun Zi 's mutualism into the doctrine embodied

The first sustained work on the subject of logic which has survived was that of Aristotle. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. [10] The formally sophisticated treatment of modern logic descends from the Greek tradition, the latter mainly being informed from the transmission of Aristotelian logic. The Organon is the name given by Aristotle 's followers the Peripatetics to the standard collection of his six works on Logic.

Logic in Islamic philosophy also contributed to the development of modern logic, which included the development of "Avicennian logic" as an alternative to Aristotelian logic. Logic ( Arabic: Mantiq) played an important role in Early Islamic philosophy. Logic ( Arabic: Mantiq) played an important role in Early Islamic philosophy. Avicenna's system of logic was responsible for the introduction of hypothetical syllogism,[11] temporal modal logic,[12][13] and inductive logic. TemplateInfobox Muslim scholars --> ( Persian /ابو علی الحسین ابن عبدالله ابن سینا (born In Logic, a hypothetical syllogism has two uses In Propositional logic it expresses a rule of inference while in the History of logic, it is a short-hand 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 A modal logic is any system of formal logic that attempts to deal with modalities. Induction or inductive reasoning, sometimes called inductive logic, is the process of Reasoning in which the premises of an argument are believed [14][15] The rise of the Asharite school, however, limited original work on logic in Islamic philosophy, though it did continue into the 15th century and had a significant influence on European logic during the Renaissance. The Ash'ari theology ( Arabic الأشاعرة al-asha`irah) is a school of early Muslim speculative theology founded by the theologian Abu al-Hasan Logic ( Arabic: Mantiq) played an important role in Early Islamic philosophy. The Renaissance (from French Renaissance, meaning "rebirth" Italian: Rinascimento, from re- "again" and nascere

In India, innovations in the scholastic school, called Nyaya, continued from ancient times into the early 18th century, though it did not survive long into the colonial period. Nyāya ( Sanskrit ni-āyá, literally "recursion" used in the sense of " Syllogism, inference" is the name given to one of the six orthodox The 18th century lasted from 1701 to 1800 in the Gregorian calendar, in accordance with the Anno Domini / Common Era numbering system The colonial era in India began in 1502 when the Portuguese established the first European trading center at Kollam In the 20th century, western philosophers like Stanislaw Schayer and Klaus Glashoff have tried to explore certain aspects of the Indian tradition of logic. The development of Indian logic can be said to date back to the anviksiki of Medhatithi Gautama (c According to Hermann Weyl (1929):

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. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician.

During the medieval period, major efforts were made to show that Aristotle's ideas were compatible with Christian faith. A Christian is a person who adheres to Christianity, a monotheistic Religion centered on the life and teachings of Jesus of Nazareth 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 logic

### Syllogistic logic

Main article: Aristotelian logic

The Organon was Aristotle's body of work on logic, with the Prior Analytics constituting the first explicit work in formal logic, introducing the syllogistic. The Organon is the name given by Aristotle 's followers the Peripatetics to the standard collection of his six works on Logic. The Organon is the name given by Aristotle 's followers the Peripatetics to the standard collection of his six works on Logic. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. Prior Analytics is Aristotle 's work on Deductive reasoning, part of his Organon, the instrument or manual of Logical The parts of syllogistic, also known by the name term logic, were the analysis of the judgements into propositions consisting of two terms that are related by one of a fixed number of relations, and the expression of inferences by means of syllogisms that consisted of two propositions sharing a common term as premise, and a conclusion which was a proposition involving the two unrelated terms from the premises. In Philosophy, term logic, also known as traditional logic, is a loose name for the way of doing logic that began with Aristotle, and that was dominant A syllogism, or logical appeal, (συλλογισμός &mdash "conclusion" "inference" (usually the categorical syllogism) is a kind of

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. Stoicism, a school of Hellenistic philosophy, was founded in Athens by Zeno of Citium in the early third century BC This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" The problem of multiple generality names a failure in Traditional logic to describe certain intuitively valid inferences Nonetheless, problems with syllogistic logic were not seen as being in need of revolutionary solutions.

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. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" In Mathematical logic, predicate logic is the generic term for symbolic Formal systems like First-order logic, Second-order logic, Many-sorted Others use Aristotle in argumentation theory to help develop and critically question argumentation schemes that are used in artificial intelligence and legal arguments. Argumentation theory, or argumentation, embraces the arts and sciences of civil debate Dialogue, conversation and persuasion studying rules of Inference Law is a system of rules enforced through a set of Institutions used as an instrument to underpin civil obedience politics economics and 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. In Mathematical logic, predicate logic is the generic term for symbolic Formal systems like First-order logic, Second-order logic, Many-sorted 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. The problem of multiple generality names a failure in Traditional logic to describe certain intuitively valid inferences 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. Quantification has two distinct meanings In Mathematics and Empirical science, it refers to human acts known as Counting and Measuring

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. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin  &ndash 26 July 1925 Analytic philosophy (sometimes analytical philosophy) is a generic term for a style of Philosophy that came to dominate English-speaking countries in the 20th century First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science Principles of Mathematical Logic is the 1950 American translation of the 1938 second edition of David Hilbert 's and Wilhelm Ackermann 's classic text David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Wilhelm Friedrich Ackermann ( March 29, 1896, Herscheid municipality Germany &ndash December 24, 1962 Lüdenscheid Year 1928 ( MCMXXVIII) was a Leap year starting on Sunday (link will display full calendar of the Gregorian calendar. 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. Alfred Tarski ( January 14, 1901, Warsaw, Russian ruled Poland – October 26, 1983, Berkeley California In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic.

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 Logic and Mathematics second-order logic is an extension of First-order logic, which itself is an extension of Propositional logic. Willard Van Orman Quine (June 25 1908 Akron, Ohio &ndash December 25 2000 (known to intimates as "Van" George Stephen Boolos ( September 4, 1940, New York City – May 27, 1996) was a Philosopher and a Mathematical logician Stewart Shapiro (born 15 June 1951) is Professor of Philosophy at the Ohio State University and

### Modal logic

Main article: Modal logic

In languages, modality deals with the phenomenon that sub-parts of a sentence may have their semantics modified by special verbs or modal particles. A modal logic is any system of formal logic that attempts to deal with modalities. 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.

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. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. Some languages distinguish between alethic moods and non-alethic moods In Logic, De Morgan's laws or De Morgan's theorem are rules in Formal logic relating pairs of dual Logical operators in a systematic manner expressed 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 axiomatizations of the alethic modalities. Clarence Irving Lewis ( April 12, 1883 Stoneham Massachusetts - February 3, 1964 Cambridge Massachusetts) usually Year 1918 ( MCMXVIII) was a Common year starting on Tuesday (link will display the full calendar of the Gregorian calendar (or a Common His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to include deontic logic and epistemic logic. Deontic logic is the field of Logic that is concerned with Obligation, Permission, and related concepts Epistemic logic is a subfield of Modal logic that is concerned with reasoning about Knowledge. 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. Arthur Norman Prior (1914 Masterton, New Zealand – 1969 Trondheim, Norway) was a noted logician. 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 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. Saul Aaron Kripke (born on November 13, 1940 in Bay Shore New York) is an American philosopher and Logician now Emeritus In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects Computational linguistics is an Interdisciplinary field dealing with the statistical and/or rule-based modeling of Natural language from a computational Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their

### 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 Reasoning which uses deductive Arguments to move from given statements ( Premises to Conclusions which must be true if the

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 (διαλεκτική is controversy the exchange of arguments and counter-arguments respectively advocating Propositions Critical thinking consists of mental processes of discernment, Analysis and Evaluation.

### 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 Logic and Mathematics with close connections to Computer science and Philosophical logic.

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. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Biography Early life Birth and family Plato was born in Athens Greece Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. Many other ancient and medieval philosophers applied mathematical ideas and methods to their philosophical claims.

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. 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 Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin  &ndash 26 July 1925 Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian [4] 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. Part of the Foundations of mathematics, Russell's paradox (also known as Russell's antinomy) discovered by Bertrand Russell in 1901 showed that the Hilbert's 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 In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most

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. Proof theory is a branch of Mathematical logic that represents proofs as formal Mathematical objects facilitating their analysis by mathematical techniques [16] 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. Gödel's completeness theorem is a fundamental theorem in Mathematical logic that establishes a correspondence between semantic truth and syntactic provability in In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models In Mathematical logic, a proof calculus corresponds to a family of Formal systems that use a common style of formal inference for its Inference rules. Thus we see how complementary the two areas of mathematical logic have been.

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. 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 (classes of mathematical structures such as groups, Fields graphs or even models 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. Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. 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 (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite In the mathematical field of Set theory, a large cardinal property is a certain kind of property of Transfinite Cardinal numbers Cardinals with such properties

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. Recursion theory, also called computability theory, is a branch of Mathematical logic that originated in the 1930s with the study of Computable functions Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone In Mathematics, the Entscheidungsproblem ( German for ' Decision problem ' is a challenge posed by David Hilbert in 1928 Alan Mathison Turing, OBE, FRS (ˈt(jʊ(ərɪŋ (23 June 1912 &ndash 7 June 1954 was an English Mathematician [17] 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. In Computational complexity theory, a complexity class is a set of problems of related complexity [18]

### Philosophical logic

Main article: Philosophical logic

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. Logic in computer science describes topics where Logic is applied to Computer science and Artificial intelligence. Alan Mathison Turing, OBE, FRS (ˈt(jʊ(ərɪŋ (23 June 1912 &ndash 7 June 1954 was an English Mathematician In Mathematics, the Entscheidungsproblem ( German for ' Decision problem ' is a challenge posed by David Hilbert in 1928 Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most The 1940s decade ran from 1940 to 1949 Events and trends The 1940s was a period between the radical 1930s and the conservative 1950s which also leads the period to be

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. See also Table of mathematical symbols Mathematical notation is used in Mathematics, and throughout the Physical sciences, Engineering 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 is in its broadest sense the use of mathematical logic for computer programming Logic programming systems such as Prolog compute the consequences of the axioms and rules in order to answer a query. Prolog is a Logic programming language It is a general purpose language often associated with Artificial intelligence and Computational linguistics

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. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Argumentation theory is one good example of how logic is being applied to artificial intelligence. Argumentation theory, or argumentation, embraces the arts and sciences of civil debate Dialogue, conversation and persuasion studying rules of Inference The ACM Computing Classification System in particular regards:

• Section F. The ACM Computing Classification System is a subject classification system for Computer science devised by the Association for Computing Machinery. 3 on Logics and meanings of programs and F. 4 on Mathematical logic and formal languages as part of the theory of computer science: this work covers formal semantics of programming languages, as well as work of formal methods such as Hoare logic
• Boolean logic as fundamental to computer hardware: particularly, the system's section B. In Theoretical computer science, formal semantics is the field concerned with the rigorous mathematical study of the meaning of Programming languages and models of In Computer science and Software engineering, formal methods are particular kind of Mathematically -based techniques for the specification, development Hoare logic (also known as Floyd&ndashHoare logic) is a Formal system developed by the British computer scientist C Boolean logic is a complete system for Logical operations It was named after George Boole, who first defined an algebraic system of 2 on Arithmetic and logic structures;
• Many fundamental logical formalisms are essential to section I. 2 on artificial intelligence, for example modal logic and default logic in Knowledge representation formalisms and methods, Horn clauses in logic programming, and description logic. A modal logic is any system of formal logic that attempts to deal with modalities. Default logic is a Non-monotonic logic proposed by Raymond Reiter to formalize reasoning with default assumptions Knowledge representation is an area in Artificial intelligence that is concerned with how to formally "think" that is how to use a symbol system to represent In Mathematical logic, a Horn clause is a clause (a Disjunction of literals with at most one positive literal Logic 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 concept definitions of an application domain (known as terminological

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. Automated theorem proving ( ATP) or automated deduction, currently the most well-developed subfield of Automated reasoning (AR is the

### 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. Argumentation theory, or argumentation, embraces the arts and sciences of civil debate Dialogue, conversation and persuasion studying rules of Inference 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. Law is a system of rules enforced through a set of Institutions used as an instrument to underpin civil obedience politics economics and society

## 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. Classical logic identifies a class of Formal logics that have been most intensively studied and most widely used In Logic, the semantic principle of bivalence states that every proposition takes exactly one of two truth values (e 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 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. Vasiliev Nicolai Alexandrovich (Николай Александрович Васильев also Vasil'ev, Vassilieff, Wassilieff (–1940 was a 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. The twentieth century of the Common Era began on Jan Łukasiewicz (ˈjan wukaˈɕɛvʲitʂ ( 21 December, 1878 &ndash 13 February, 1956) was a Polish Mathematician born A ternary, three-valued or trivalent logic (sometimes abbreviated 3VL) is a term to describe any of several Multi-valued logic systems in which Multi-valued logics are logical calculi in which there are more than two Truth values Traditionally logical calculi are two-valued—that is there are only two possible

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. Fuzzy logic is a form of Multi-valued logic derived from Fuzzy set theory to deal with Reasoning that is approximate rather than precise In Mathematics, the real numbers may be described informally in several different ways

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. Intuitionistic logic, or constructivist logic, is the Symbolic logic system originally developed by Arend Heyting to provide a formal basis for Brouwer Luitzen Egbertus Jan Brouwer ɛxˈbɛʁtəs jɑn ˈbʁʌuəʁ ( February 27 1881, Overschie – December 2 1966, Blaricum This article uses forms of logical notation For a concise description of the symbols used in this notation see Table of logic symbols. In the Philosophy of mathematics, intuitionism, or neointuitionism (opposed to Preintuitionism) is an approach to Mathematics as the constructive Brouwer rejected formalisation in mathematics, but his student Arend Heyting studied intuitionistic logic formally, as did Gerhard Gentzen. Arend Heyting ( May 9, 1898 &ndash July 9, 1980) was a Dutch Mathematician and Logician. Gerhard Karl Erich Gentzen ( November 24, 1909, Greifswald, Germany &ndash August 4, 1945, Prague, Czechoslovakia 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. In the Philosophy of mathematics

Modal logic is not truth conditional, and so it has often been proposed as a non-classical logic. A modal logic is any system of formal logic that attempts to deal with modalities. However, modal logic is normally formalised with the principle of the excluded middle, and its relational semantics is bivalent, so this inclusion is disputable. Kripke semantics (also known as relational semantics or frame semantics, and often confused with Possible world semantics) is a formal Semantics On the other hand, modal logic can be used to encode non-classical logics, such as intuitionistic logic.

Bayesian probability can be interpreted as a system of logic where probability is the subjective truth value. Bayesian probability interprets the concept of Probability as 'a measure of a state of knowledge'.

### Implication: strict or material?

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 Principle of explosion, a law of Classical logic stating that inconsistent premises always

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. The principle of explosion is the law of Classical logic and a few other systems (e 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. Clarence Irving Lewis ( April 12, 1883 Stoneham Massachusetts - February 3, 1964 Cambridge Massachusetts) usually 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 a kind of non-classical logic requiring the antecedent and consequent of implications be relevantly related

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. The philosopher Paul Grice proposed four conversational maxims that arise from the Pragmatics of natural Language. Monotonicity of Entailment is a property of many Logical systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions

### 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. A paraconsistent logic is a Logical system that attempts to deal with Contradictions in a discriminating way 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. Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications be relevantly related A paraconsistent logic is a Logical system that attempts to deal with Contradictions in a discriminating way 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 Symbolic logic system originally developed by Arend Heyting to provide a formal basis for Brouwer The principle of explosion is the law of Classical logic and a few other systems (e Graham Priest, the main proponent of dialetheism, has argued for paraconsistency on the grounds that there are in fact, true contradictions. Graham Priest (born 1948, London) is Boyce Gibson Professor of Philosophy at the University of Melbourne and a regular visitor at St Dialetheism is the view that there are true contradictions or dialetheias [19]

### 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?"[20] 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. " Is logic empirical? " is the title of two articles that discuss the idea that the algebraic properties of logic may or should be empirically determined in particular they Epistemology (from Greek επιστήμη - episteme, "knowledge" + λόγος, " Logos " or theory of knowledge The laws of Classical logic are a small collection of fundamental sentences of Propositional logic and Boolean algebra, from which may be derived 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 Willard Van Orman Quine (June 25 1908 Akron, Ohio &ndash December 25 2000 (known to intimates as "Van" Mechanics ( Greek) is the branch of Physics concerned with the behaviour of physical bodies when subjected to Forces or displacements General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Contemporary philosophical realism is the belief in a Reality that is completely Ontologically independent of our conceptual schemes linguistic practices beliefs The principle of distributivity states that the algebraic Distributive law is valid for Classical logic, where both Logical conjunction and In Mathematical physics and Quantum mechanics, quantum logic is a set of rules for Reasoning about propositions which takes the principles of Garrett Birkhoff ( January 19, 1911, Princeton, New Jersey, USA – November [21]

Another paper by the same name by Sir Michael Dummett argues that Putnam's desire for realism mandates the law of distributivity. Sir Michael Anthony Eardley Dummett FBA DLitt (born 1925 is a leading British Philosopher. [22] 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. Metaphysics is the branch of Philosophy investigating principles of reality transcending those of any particular science In Philosophy, the term anti-realism is used to describe anyposition involving either the denial of an objective Reality of Entities of a certain

## Notes

1. ^ Logikos, Henry George Liddell, Robert Scott, A Greek-English Lexicon, at Perseus
2. ^ Online Etymology Dictionary
3. ^ J. Robert Cox and Charles Arthur Willard, eds. Advances in Argumentation Theory and Research, Southern Illinois University Press, 1983 ISBN 0809310503, ISBN 978-0809310500
4. ^ a b c Alfred North Whitehead and Bertrand Russell, Principia Mathematical to *56, Cambridge University Press, 1967, ISBN 0-521-62606-4
5. ^ Plato, The Portable Plato, edited by Scott Buchanan, Penguin, 1976, ISBN 0-14-015040-4
6. ^ Aristotle, The Basic Works, Richard Mckeon, editor, Modern Library, 2001, ISBN 0-375-75799-6, see especially, Posterior Analytics. The Posterior Analytics is a text from Aristotle 's Organon that deals with demonstration, Definition, and Scientific knowledge
7. ^ a b For a more modern treatment, see A. G. Hamilton, Logic for Mathematicians, Cambridge, 1980, ISBN 0-521-29291-3
8. ^ S. Kak (2004). Subhash Kak (सुभाष काक Subhāṣ Kāk) (born March 26, 1947 in Srinagar, Kashmir) is an Indian American The Architecture of Knowledge. CSC, Delhi.
9. ^ McGreal 1995, p. 33
10. ^ 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. "
11. ^ Lenn Evan Goodman (2003), Islamic Humanism, p. 155, Oxford University Press, ISBN 0195135806.
12. ^ History of logic: Arabic logic, Encyclopædia Britannica. The Encyclopædia Britannica is a general English-language encyclopaedia published by Encyclopædia Britannica Inc
13. ^ Dr. Lotfollah Nabavi, Sohrevardi's Theory of Decisive Necessity and kripke's QSS System, Journal of Faculty of Literature and Human Sciences.
14. ^ Science and Muslim Scientists, Islam Herald.
15. ^ Wael B. Hallaq (1993), Ibn Taymiyya Against the Greek Logicians, p. 48. Oxford University Press, ISBN 0198240430.
16. ^ Mendelson, "Formal Number Theory: Gödel's Incompleteness Theorem"
17. ^ Brookshear, "Computability: Foundations of Recursive Function Theory"
18. ^ Brookshear, "Complexity"
19. ^ Priest, Graham (2004), "Dialetheism", Stanford Encyclopedia of Philosophy, Edward N. Graham Priest (born 1948, London) is Boyce Gibson Professor of Philosophy at the University of Melbourne and a regular visitor at St Zalta (ed. ), http://plato.stanford.edu/entries/dialetheism.
20. ^ Putnam, H. (1969), "Is Logic Empirical?", Boston Studies in the Philosophy of Science. 5.
21. ^ Birkhoff, G. , and von Neumann, J. (1936), "The Logic of Quantum Mechanics", Annals of Mathematics 37, 823–843. The Annals of Mathematics (ISSN 0003-486X abbreviated as Ann of Math
22. ^ Dummett, M. (1978), "Is Logic Empirical?", Truth and Other Enigmas. ISBN 0-674-91076-1

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