Logic
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Logic, from Classical Greek λόγος (logos), originally meaning the word, but also referring to speech or reason is the science that evaluates reasoning within arguments.
Nature of logicEdit
Logic is generally understood and accepted as a set of rules that tell us when an argument's premises support their conclusion. An understanding of just what logic is, can be enhanced by delineating it from what it is not:
 Logic is not the 'groundness of being'  that's metaphysics .
 Logic is not a set of laws that governs the universe  that's physics .
 Logic is not an immaterial "entity" that transcends reality  that's speculative theology .
 Logic is not a set of laws that governs human behavior  that's psychology .
 Logic is not a method for 'studying the world'  that's science .
 Logic is not the method for assessing axioms  that's a matter of pure reason .
 Logic is not a way of evaluating 'truth'  that's philosophy.
Arguments Edit
An argument is made up of groups of statements we call propositions  and every proposition contains a truth value. In the case of classic logic, this truth value is either "true" or false"  a rule we can call "bivalence". Formally: for any proposition P, P is either true or false. We use the term "proposition" instead of "sentence" because a proposition is more than just a sentence, it is a declarative sentence  asserting a specific and coherent claim, that is either true or false.
There are two types of propositions within an argument. The first, the premise, makes a commitment to truth, and is used as evidence to support the second type of proposition, the conclusion, which is the claim the argument is supposed to prove. An argument must at least imply one of each.
Here is an example:
 Premise 1: Either that's a gun in your pocket, or you're glad to see me.
 Premise 2: Uh... that's not a gun in your pocket.
 Conclusion: So you must be glad to see me.
The study of logic, therefore, is the effort to determine the conditions under which one is justified in passing from the premises to the conclusion that logically must follow them. When an argument has this sort of justification, it is a valid argument. A valid argument with true premises is a sound argument, and guarantees the truth of its conclusion.
Informal, formal, and symbolic logicEdit
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.
 Formal logic is the study of inference with purely formal content, where that content is made explicit. (An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. 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.)
 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".
While formal logic is old, dating back more than two millennia, most of symbolic logic is comparatively new, and arises with the application of insights from mathematics to problems in logic. Generally, a symbolic logic is captured by a formal system, comprising a formal language including rules for creating expressions in the language, and a set of rules of derivation. The expressions 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.
For example, consider a very simple formal system that has just the symbols "p", "q", and "and" in its language. Its only rules for creating expressions are (1) "'p and q' and 'q and p' are expressions" and (2) Any expression compounded with another by 'and" is also an expression". Its only rule of derivation is "from any expression of the form 'p and q', you may conclude 'p'". The intended interpretation of the "p" and "q" is that they stand for any sentence. The intended interpretation of 'and' is expressed by specifying when sentences that contain 'and' are true. Most systems would interpret 'and' like this: sentences containing 'and' are true only when the expressions on either side of it (both of them) are true.
A formal system can also have axioms. An axiom is a sentence that counts as always true within the system. For example, many systems have as an axiom the sentence "If P implies Q and P is the case, then Q is the case." To go along with the axioms the system will have a special rule of derivation, called the 'rule of substitution'. It says that you can derive from any axiom a sentence that is just like it, except that other sentences have been substituted for the 'P' and the 'Q'. For example, from the axiom above, we can conclude the following: "If R&S implies that T or U, and R&S is the case; then it is the case that T or U." (This assumes that "R&S" and "T or U" are expressions in the formal system.) Most formal systems have either a rich set of rules of derivation, but few or no axioms; or a rich set of axioms but only the derivation rule of substitution.
Sentences that are derived using the system's axioms and rules of derivation are called theorems.
Consistency, soundness, and completenessEdit
There are three valuable properties that formal systems can have:
 Consistency, which means that none of the theorems of the system contradict each other.
 Soundness, which means that the system's rules of derivation will never let you infer anything false, so long as you start with only true premises. So if a system is sound (and its axioms, if any, are true), then the theorems of a sound formal system are the truths. All of the theorems of a system that has no axioms are its truths and sometimes the truths of such a system are called 'logical truths.' (Note that if a system is not consistent, it cannot be sound. This is because a contradiction is always false, so if two theorems contradict at least one is false.)
 Completeness, which means that there are no true sentences in the system that cannot, at least in principle, be proved using the derivation rules (and axioms, if any) of the system.
Not all systems achieve all three virtues. It has been proven by Kurt Gödel that a system with enough axioms and/or rules of derivation to derive the principles of arithmetic cannot be both consistent and complete. This is called Gödel's Incompleteness Theorem.
Important families of formal systemsEdit
Formal logic encompasses a wide variety of logical systems. Various systems of logic discussed below include term logic, predicate logic, propositional 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.
Rival conceptions of logicEdit
Logic arose (see below) from a concern with correctness of argumentation. The conception of logic as the study of argument is historically fundamental, and was how the founders of distinct traditions of logic, namely Plato, Aristotle, Mozi, Aksapada Gautama, Farabi, and Avicenna, 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).
By contrast Immanuel Kant introduced an alternative idea as to what logic is. He argued that logic should be conceived as the science of judgement, an idea taken up in Gottlob Frege's logical and philosophical work, where thought (German: Gedanke) is substituted for judgement (German: Urteil). On this conception, the valid inferences of logic follow from the structural features of judgements or thoughts.
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.
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.
Relation to other sciencesEdit
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 (i.e. it describes how reasoning ought to take place), whereas psychology is descriptive, so the overlap is not so marked. Gottlob Frege, however, was adamant about antipsychologism: that logic should be understood in a manner independent of the idiosyncrasies of how particular people might reason.
Deductive and inductive reasoningEdit
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 wellunderstood notions of semantics. Inductive validity on the other hand requires us to define a reliable generalization of some set of observations. The task of providing this definition may be approached in various ways, some less formal than others; some of these definitions may use mathematical models of probability. For the most part this discussion of logic deals only with deductive logic.
HistoryEdit
 Main article: History of logic
Many ancient civilizations developed systems of argumentation and studied logical paradoxes. Logic was studied in several ancient civilizations, including India,^{[1]} China,^{[2]} Babylonia, Persia, and Greece.
While many cultures have employed intricate systems of reasoning and math, logic as an explicit analysis of the methods of reasoning received sustained development originally in three places: India in the 6th century BC, China in the 5th century BC, and Greece between the 4th century BC and the 1st century BC.
In the Eastern world, logic was developed by Buddhists and Jains, with Indian logic and Chinese logic. 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 China, the tradition of scholarly investigation into logic was repressed by the Qin dynasty following the legalist philosophy of Han Feizi. In subsequent centuries, China eventually adopted forms of Indian logic.
The Greek tradition mainly comes from the transmission of Aristotelian logic, which probably developed independently of Indian logic, and further development by Islamic logicians and then Medieval logicians.
In the Islamic world, Logic was further extended by AlFarabi who categorized it into two separate groups (idea and proof). Later, Avicenna revived the study of logic and developed relationship between temporalis and the implication. Averroes defined logic as "the tool for distinguishing between the true and the false";^{[3]} Richard Whately, '"the Science, as well as the Art, of reasoning"; and Frege, "the science of the most general laws of truth".
During the medieval period, after it was shown that Aristotle's ideas were largely compatible with faith, a greater emphasis was placed upon Aristotle's logic. 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.
The formally sophisticated treatment of modern logic descends from the Greek, Islamic and Indian traditions. For example, the pioneers of Boolean logic were aware of Indian logic (Ganeri 2001).
The late 18th to early 19th centuries saw the development of symbolic logic and mathematical logic introduced new paradoxes and more powerful techniques to deal with them.
In the 20th century, Western philosophers like Stanislaw Schayer and Klaus Glashoff have tried to explore certain aspects of the Indian tradition of logic.
History of logicEdit
 Main article: History of logic
Several ancient civilizations have employed intricate systems of reasoning and asked questions about logic or propounded logical paradoxes. In Indian logic, the Nasadiya Sukta of the Rigveda (RV 10.129) contains ontological speculation in terms of various logical divisions that were later recast formally as the four circles of catuṣkoṭi: "A", "not A", "Neither A or not A", and "Both not A and not not A".^{[4]} The Chinese logical philosopher Gongsun Long (ca. 325–250 BC) proposed the paradox "One and one cannot become two, since neither becomes two."^{[5]} Also, the Chinese School of Names is recorded as having examined logical puzzles such as "A White Horse is not a Horse" as early as the fifth century BCE.^{[6]} In China, the tradition of scholarly investigation into logic, however, was repressed by the Qin dynasty following the legalist philosophy of Han Feizi. The earliest sustained Greek work on the subject of logic is that of Aristotle.^{[7]} Aristotelian logic became widely accepted in science and mathematics and remained in wide use in Europe until the early 19th century.
Islamic logic contributed to the development of modern logic, which included the development of "Avicennian logic"^{[8]} as an alternative to Aristotelian logic. Avicenna's system of logic was responsible for the introduction of hypothetical syllogism,^{[9]} temporal modal logic,^{[10]}^{[11]} and inductive logic.^{[12]}^{[13]} It later had a significant influence on European logic during the Renaissance. In Europe during the later medieval period, major efforts were made to show that Aristotle's ideas were compatible with Christian faith. During the later period of the Middle Ages, logic became a main focus of philosophers, who would engage in critical logical analyses of philosophical arguments.
In India, innovations in the scholastic school, called Nyaya, continued from ancient times into the early 18th century with the NavyaNyaya school. By the 16th century, it developed theories resembling modern logic, such as Gottlob Frege's "distinction between sense and reference of proper names" and his "definition of number," as well as the theory of "restrictive conditions for universals" anticipating some of the developments in modern set theory.^{[14]} Since 1824, Indian logic attracted the attention of many Western scholars, and has had an influence on important 19thcentury logicians such as Charles Babbage, Augustus De Morgan, and George Boole.^{[15]} In the 20th century, Western philosophers like Stanislaw Schayer and Klaus Glashoff have explored Indian logic more extensively.
The syllogistic logic developed by Aristotle predominated in the West until the mid19th century, when interest in the foundations of mathematics stimulated the development of symbolic logic (now called mathematical logic). In 1854, George Boole published An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, introducing symbolic logic and the principles of what is now known as Boolean logic. In 1879, Gottlob Frege published Begriffsschrift which inaugurated modern logic with the invention of quantifier notation. From 1910 to 1913, Alfred North Whitehead and Bertrand Russell published Principia Mathematica^{[16]} on the foundations of mathematics, attempting to derive mathematical truths from axioms and inference rules in symbolic logic. In 1931, Gödel raised serious problems with the foundationalist program and logic ceased to focus on such issues.
The development of logic since Frege, Russell and Wittgenstein had a profound influence on the practice of philosophy and the perceived nature of philosophical problems (see Analytic philosophy), and Philosophy of mathematics. Logic, especially sentential logic, is implemented in computer logic circuits and is fundamental to computer science. Logic is commonly taught by university philosophy departments often as a compulsory discipline.
Topics in logicEdit
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.
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 logicEdit
 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 parts of syllogistic, also known by the name yisemral
Edit
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.
Aristotle's work was regarded in classical times and from medieval times in Europe and the Middle East as the very picture of a fully worked out system. It was not alone: the Stoics proposed a system of propositional logic that was studied by medieval logicians; nor was the perfection of Aristotle's system undisputed; for example the problem of multiple generality was recognised in medieval times. Nonetheless, problems with syllogistic logic were not seen as being in need of revolutionary solutions with yisemral. Today, some academics claim that Aristotle's system is generally seen as having little more than historical value (though there is some current interest in extending term logics), regarded as made obsolete by the advent of sentential logic and the predicate calculus. Others use Aristotle in argumentation theory to help develop and critically question argumentation schemes that are used in artificial intelligence and [[le
Predicate logicEdit
 Main article: Predicate logic
Logic as it is studied today is a very different subject to that studied before, and the principal difference is the innovation of predicate logic. Whereas Aristotelian syllogistic logic specified the forms that the relevant 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 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 firstorder 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.
Frege's original system of predicate logic was not first, but secondorder. Secondorder logic is most prominently defended (against the criticism of Willard Van Orman Quine and others) by George Boolos and Stewart Shapiro.
Modal logicEdit
 Main article: Modal logic
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.
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 graphtheoretic way of looking at modality that has driven many applications in computational linguistics and computer science, such as dynamic logic.
==
 Main article: Deductive reasoning==
Mathematical logicEdit
 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.
The earliest use of math and geometry in relation to logic and philosophy goes back to the ancient Greeks such as Euclid, Plato, and Aristotle. Many other ancient and medieval philosophers applied mathematical ideas and methods to their philosophical claims.
The boldest attempt to apply logic to mathematics was undoubtedly the logicism pioneered by philosopherlogicians 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.
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 firstorder 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.
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.
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 ChurchTuring 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.
Philosophical logicEdit
 Main article: Philosophical logic
Philosophical logic deals with formal descriptions of natural language. Most philosophers assume that the bulk of "normal" proper reasoning can be captured by logic, if one can find the right method for translating ordinary language into that logic. Philosophical logic is essentially a continuation of the traditional discipline that was called "Logic" before it was supplanted by the invention of mathematical logic. Philosophical logic has a much greater concern with the connection between natural language and logic. As a result, philosophical logicians have contributed a great deal to the development of nonstandard logics (e.g., free logics, tense logics) as well as various extensions of classical logic (e.g., modal logics), and nonstandard semantics for such logics (e.g., Kripke's technique of supervaluations in the semantics of 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.
Philosophy of language underwent a renaissance in the 20th century because of the work of Ludwig Wittgenstein.
Logic and computationEdit
 Main article: Logic in computer science
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.
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.
Today, logic is extensively applied in the fields of artificial intelligence, and computer science, and these fields provide a rich source of problems in formal and informal logic. Argumentation theory is one good example of how logic is being applied to artifical intelligence. The ACM Computing Classification System in particular regards:
 Section F.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.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.
Furthermore, computers can be used as tools for logicians. For example, in symbolic logic and mathematical logic, proofs by humans can be computerassisted. 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.
Argumentation TheoryEdit
Argumentation theory is the study and research of informal logic, fallacies, and critical questions as they relate to every day and practical situations. Specific types of dialogue can be analyzed and questioned to reveal premises, conclusions, and fallacies. Argumentation theory is now applied in artificial intelligence and law.
Controversies in logicEdit
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 middleEdit
 Main article: Classical logic
The logics discussed above are all "bivalent" or "twovalued"; that is, they are most naturally understood as dividing propositions into the true and the false propositions. Systems which reject bivalence are known as nonclassical logics.
In 1910 Nicolai A. Vasiliev rejected the law of excluded middle and the law of contradiction and proposed the law of excluded fourth and logic tolerant to contradiction. In the early 20th century Jan Łukasiewicz investigated the extension of the traditional true/false values to include a third value, "possible", so inventing ternary logic, the first multivalued logic.
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.
Modal logic is not truth conditional, and so it has often been proposed as a nonclassical logic. However, modal logic is normally formalised with the principle of the excluded middle, and its relational semantics is bivalent, so this inclusion is disputable. On the other hand, modal logic can be used to encode nonclassical logics, such as intuitionistic logic.
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.
Implication: strict or material?Edit
 Main article: Paradox of entailment
It is obvious that the notion of implication formalised in classical logic does not comfortably translate into natural language by means of "if... then...", due to a number of problems called the paradoxes of material implication.
The first class of paradoxes involves counterfactuals, such as "If the moon is made of green cheese, then 2+2=5", which are puzzling because natural language does not support the principle of explosion. Eliminating this class of paradoxes was the reason for C. I. Lewis's formulation of strict implication, which eventually led to more radically revisionist logics such as relevance logic.
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.
Tolerating the impossibleEdit
 Main article: Paraconsistent logic
Closely related to questions arising from the paradoxes of implication comes the radical suggestion that logic ought to tolerate inconsistency. Relevance logic and paraconsistent logic are the most important approaches here, though the concerns are different: a key consequence of classical logic and some of its rivals, such as intuitionistic logic, is that they respect the principle of explosion, which means that the logic collapses if it is capable of deriving a contradiction. Graham Priest, the main proponent of dialetheism, has argued for paraconsistency on the striking grounds that there are in fact, true contradictions (Priest 2004).
Is logic empirical?Edit
 Main article: Is logic empirical?
What is the epistemological status of the laws of logic? What sort of arguments is appropriate for criticising purported principles of logic? In an influential paper entitled Is logic empirical? 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.
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 antirealism.
ReferencesEdit
 Birkhoff, G., and von Neumann, J. (1936), "The Logic of Quantum Mechanics", Annals of Mathematics 37, 823–843.
 Banda, Obdulio (1944), "Hacia el análisis ontológicocategorial de lo lógico", IIPCIAL, Fondo Editorial (2008).ISBN 9789972998225
 Cohen, R.S, and Wartofsky, M.W. (1974), Logical and Epistemological Studies in Contemporary Physics, Boston Studies in the Philosophy of Science, D. Reidel Publishing Company, Dordrecht, Netherlands. ISBN 9027703779.
 Finkelstein, D. (1969), "Matter, Space, and Logic", in R.S. Cohen and M.W. Wartofsky (eds. 1974).
 Gabbay, D.M., and Guenthner, F. (eds., 20012005), Handbook of Philosophical Logic, 13 vols., 2nd edition, Kluwer Publishers, Dordrecht.
 Vincent F. Hendricks, Thought 2 Talk: A Crash Course in Reflection and Expression, New York: Automatic Press / VIP, 2005, ISBN 8799101378
 Hilbert, D., and Ackermann, W. (1928), Grundzüge der theoretischen Logik (Principles of Theoretical Logic), SpringerVerlag.
 Hodges, W. (2001), Logic. An introduction to Elementary Logic, Penguin Books.
 Hofweber, T. (2004), "Logic and Ontology", Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), Eprint.
 Hughes, R.I.G. (ed., 1993), A Philosophical Companion to FirstOrder Logic, Hackett Publishing.
 Kneale, William, and Kneale, Martha, (1962), The Development of Logic, Oxford University Press, London, UK.
 Priest, Graham (2004), "Dialetheism", Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), Eprint.
 Putnam, H. (1969), "Is Logic Empirical?", Boston Studies in the Philosophy of Science' 5.
 Smith, B. (1989), "Logic and the Sachverhalt", The Monist 72(1), 52–69.
See alsoEdit

NotesEdit
 ↑ For example, Nyaya (syllogistic recursion) dates back 1900 years.
 ↑ Mohists and the school of Names date back at 2200 years.
 ↑ Averroes, In Arist. Physicam I, textus 35, ed. Juntina, IV, fol. 11vb.
 ↑ Kak, S. (2004). The Architecture of Knowledge. Delhi: CSC.
 ↑ The four Catuṣkoṭi logical divisions are formally very close to the four opposed propositions of the Greek tetralemma, which in turn are analogous to the four truth values of modern relevance logic Cf. Belnap (1977); Jayatilleke, K. N., (1967, The logic of four alternatives, in Philosophy East and West, University of Hawaii Press).
 ↑ "School of Names". Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/schoolnames/. Retrieved 5 September 2008.
 ↑ E.g., Kline (1972, p.53) wrote "A major achievement of Aristotle was the founding of the science of logic".
 ↑ Goodman, Lenn Evan (1992). Avicenna. Routledge. p. 184. ISBN 9780415019293.
 ↑ Goodman, Lenn Evan (2003). Islamic Humanism. Oxford University Press. p. 155. ISBN 0195135806.
 ↑ "History of logic: Arabic logic". Encyclopædia Britannica. http://www.britannica.com/EBchecked/topic/346217/historyoflogic/65928/Arabiclogic.
 ↑ Nabavi, Lotfollah. "Sohrevardi's Theory of Decisive Necessity and kripke's QSS System". Journal of Faculty of Literature and Human Sciences. Archived from the original on 26 January 2008. http://web.archive.org/web/20080126100838/http://public.ut.ac.ir/html/fac/lit/articles.html.
 ↑ "Science and Muslim Scientists". Islam Herald. Archived from the original on 17 December 2007. http://web.archive.org/web/20071217150016/http://www.islamherald.com/asp/explore/science/science_muslim_scientists.asp.
 ↑ Hallaq, Wael B. (1993). Ibn Taymiyya Against the Greek Logicians. Oxford University Press. p. 48. ISBN 0198240430.
 ↑ Kisor Kumar Chakrabarti (June 1976), "Some Comparisons Between Frege's Logic and NavyaNyaya Logic", Philosophy and Phenomenological Research (International Phenomenological Society) 36 (4): 554563, http://www.jstor.org/stable/2106873, "This paper consists of three parts. The first part deals with Frege's distinction between sense and reference of proper names and a similar distinction in NavyaNyaya logic. In the second part we have compared Frege's definition of number to the NavyaNyaya definition of number. In the third part we have shown how the study of the socalled 'restrictive conditions for universals' in NavyaNyaya logic anticipated some of the developments of modern set theory."
 ↑ Jonardon Ganeri (2001), Indian logic: a reader, Routledge, pp. vii, 5, 7, ISBN 0700713069
 ↑ Whitehead, Alfred North; Russell, Bertrand (1967). Principia Mathematica to *56. Cambridge University Press. ISBN 0521626064.
ReferencesEdit
 Nuel Belnap, (1977). A useful fourvalued logic. In Dunn & Eppstein, Modern uses of multiplevalued logic. Reidel: Boston.
 Brookshear, J. Glenn (1989). Theory of computation: formal languages, automata, and complexity. Redwood City, Calif.: Benjamin/Cummings Pub. Co.. ISBN 0805301437.
 Cohen, R.S, and Wartofsky, M.W. (1974). Logical and Epistemological Studies in Contemporary Physics. Boston Studies in the Philosophy of Science. D. Reidel Publishing Company: Dordrecht, Netherlands. ISBN 9027703779.
 Finkelstein, D. (1969). "Matter, Space, and Logic". in R.S. Cohen and M.W. Wartofsky (eds. 1974).
 Gabbay, D.M., and Guenthner, F. (eds., 2001–2005). Handbook of Philosophical Logic. 13 vols., 2nd edition. Kluwer Publishers: Dordrecht.
 Hilbert, D., and Ackermann, W, (1928). Grundzüge der theoretischen Logik (Principles of Mathematical Logic). SpringerVerlag. OCLC 2085765
 Susan Haack, (1996). Deviant Logic, Fuzzy Logic: Beyond the Formalism, University of Chicago Press.
 Hodges, W., (2001). Logic. An introduction to Elementary Logic, Penguin Books.
 Hofweber, T., (2004), Logic and Ontology. Stanford Encyclopedia of Philosophy. Edward N. Zalta (ed.).
 Hughes, R.I.G., (1993, ed.). A Philosophical Companion to FirstOrder Logic. Hackett Publishing.
 Kline, Morris (1972). Mathematical Thought From Ancient to Modern Times. Oxford University Press. ISBN 0195061357.
 Kneale, William, and Kneale, Martha, (1962). The Development of Logic. Oxford University Press, London, UK.
 Liddell, Henry George; Scott, Robert. "Logikos". A GreekEnglish Lexicon. Perseus Project. http://www.perseus.tufts.edu/cgibin/ptext?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3D%2363716. Retrieved 8 May 2009.
 Mendelson, Elliott, (1964). Introduction to Mathematical Logic. Wadsworth & Brooks/Cole Advanced Books & Software: Monterey, Calif. OCLC 13580200
 Harper, Robert (2001). "Logic". Online Etymology Dictionary. http://www.etymonline.com/index.php?term=logic. Retrieved 8 May 2009.
 Smith, B., (1989). "Logic and the Sachverhalt". The Monist 72(1):52–69.
 Whitehead, Alfred North and Bertrand Russell, (1910). Principia Mathematica. Cambridge University Press: Cambridge, England. OCLC 1041146
External links and further readingEdit
Find more about Logic at Wikipedia's sister projects  
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 LogicWiki, an external wiki
 Logic at Memory Alpha (a Star Trek wiki)
 Introductions and tutorials
 An Introduction to Philosophical Logic, by Paul Newall, aimed at beginners
 forall x: an introduction to formal logic, by P.D. Magnus, covers sentential and quantified logic
 Logic SelfTaught: A Workbook (originally prepared for online logic instruction).
 Nicolas Rescher. (1964). Introduction to Logic, St. Martin's Press.
 Translation Tips, by Peter Suber, for translating from English into logical notation
 Math & Logic: The history of formal mathematical, logical, linguistic and methodological ideas. In The Dictionary of the History of Ideas.
 Essays
 "Symbolic Logic" and "The Game of Logic", Lewis Carroll, 1896.
 Math & Logic: The history of formal mathematical, logical, linguistic and methodological ideas. In The Dictionary of the History of Ideas.
 Reference material
 Translation Tips, by Peter Suber, for translating from English into logical notation.
 Ontology and History of Logic in Western Thought. An Introduction Annotated bibliography
 Reading lists
 The London Philosophy Study Guide offers many suggestions on what to read, depending on the student's familiarity with the subject:

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