Logic (from the Greek logikē) is the study of arguments. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking. In philosophy, the study of logic figures in most major areas of focus: epistemology, ethics, metaphysics. In mathematics, it is the study of valid inferences within some formal language. Richard Whately, "the Science, as well as the Art, of reasoning;" and Gottlob Frege, "the science of the most general laws of truth." The article Definitions of logic provides citations for these and other definitions.

Logic is often divided into two parts, inductive reasoning and deductive reasoning. The first is drawing general conclusions from specific examples, the second drawing logical conclusions from definitions and axioms. A similar dichotomy, used by Aristotle, is analysis and synthesis. Here the first takes an object of study and examines its component parts. The second considers how parts can be combined to form a whole.

Logic is also studied in argumentation theory.

Nature

The concept of logical form is central to logic, it being held that the validity of an argument is determined by its logical form, not by its content. Traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logics.

* Informal logic is the study of natural language arguments. The study of fallacies is an especially important branch of informal logic. The dialogues of Plato are good examples of informal logic. Formal logic is the study of inference with purely formal content. 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. Modern formal logic follows and expands on Aristotle. 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 often divided into two branches: propositional logic and predicate logic.

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.

Logical form

Logic is generally accepted to be formal, in that it aims to analyze and represent the form (or logical form) of any valid argument type. The form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. If one considers the notion of form to be too philosophically loaded, one could say that formalizing is nothing else than translating English sentences in the language of logic.

This is known as showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a considerable variety of form and complexity that makes their use in inference impractical. It requires, first, ignoring those grammatical features which are irrelevant to logic (such as gender and declension if the argument is in Latin), replacing conjunctions which are not relevant to logic (such as 'but') with logical conjunctions like 'and' and replacing ambiguous or alternative logical expressions ('any', 'every', etc.) with expressions of a standard type (such as 'all', or the universal quantifier ∀).

Second, certain parts of the sentence must be replaced with schematic letters. Thus, for example, the expression 'all As are Bs' shows the logical form which is common to the sentences 'all men are mortals', 'all cats are carnivores', 'all Greeks are philosophers' and so on.

That the concept of form is fundamental to logic was already recognized in ancient times. Aristotle uses variable letters to represent valid inferences in Prior Analytics, leading Jan Łukasiewicz to say that the introduction of variables was 'one of Aristotle's greatest inventions'. According to the followers of Aristotle (such as Ammonius), only the logical principles stated in schematic terms belong to logic, and not those given in concrete terms. The concrete terms 'man', 'mortal', etc., are analogous to the substitution values of the schematic placeholders 'A', 'B', 'C', which were called the 'matter' (Greek 'hyle') of the inference.

The fundamental difference between modern formal logic and traditional or Aristotelian logic lies in their differing analysis of the logical form of the sentences they treat.

* In the traditional view, the form of the sentence consists of (1) a subject (e.g. 'man') plus a sign of quantity ('all' or 'some' or 'no'); (2) the copula which is of the form 'is' or 'is not'; (3) a predicate (e.g. 'mortal'). Thus: all men are mortal. The logical constants such as 'all', 'no' and so on, plus sentential connectives such as 'and' and 'or' were called 'syncategorematic' terms (from the Greek 'kategorei' – to predicate, and 'syn' – together with). This is a fixed scheme, where each judgement has an identified quantity and copula, determining the logical form of the sentence.

According to the modern view, the fundamental form of a simple sentence is given by a recursive schema, involving logical connectives, such as a quantifier with its bound variable, which are joined to by juxtaposition to other sentences, which in turn may have logical structure.

The modern view is more complex, since a single judgement of Aristotle's system will involve two or more logical connectives. For example, the sentence "All men are mortal" involves in term logic two non-logical terms "is a man" (here M) and "is mortal" (here D): the sentence is given by the judgement A(M,D). In predicate logic the sentence involves the same two non-logical concepts, here analyzed as $m(x)$ and $d(x)$, and the sentence is given by $\backslash forall\; x.\; (m(x)\; \backslash rightarrow\; d(x))$, involving the logical connectives for universal quantification and implication.

But equally, the modern view is more powerful: medieval logicians recognized the problem of multiple generality, where Aristotelean logic is unable to satisfactorily render such sentences as "Some guys have all the luck", because both quantities "all" and "some" may be relevant in an inference, but the fixed scheme that Aristotle used allows only one to govern the inference. Just as linguists recognize recursive structure in natural languages, it appears that logic needs recursive structure.

Deductive and inductive reasoning

Deductive reasoning concerns what follows necessarily from given premises (if a, then b). However, inductive reasoning—the process of deriving a reliable generalization from observations—has sometimes been included in the study of logic. Correspondingly, we must distinguish between deductive validity and inductive validity (called "cogency"). An inference is deductively valid if and only if there is no possible situation in which all the premises are true but the conclusion false. An inductive argument can be neither valid nor invalid; its premises give only some degree of probability, but not certainty, to its conclusion.

The notion of deductive validity can be rigorously stated for systems of formal logic in terms of the well-understood notions of semantics. Inductive validity on the other hand requires us to define a reliable generalization of some set of observations. The task of providing this definition may be approached in various ways, some less formal than others; some of these definitions may use mathematical models of probability. For the most part this discussion of logic deals only with deductive logic.

Consistency, validity, soundness, and completeness

Among the important properties that logical systems can have: Consistency, which means that no theorem of the system contradicts another. Validity, which means that the system's rules of proof will never allow a false inference from true premises. A logical system has the property of soundness when the logical system has the property of validity and only uses premises that prove true (or, in the case of axioms, are true by definition).

Rival conceptions of logic

Logic arose (see below) from a concern with correctness of argumentation. Modern logicians usually wish to ensure that logic studies just those arguments that arise from appropriately general forms of inference. For example, Thomas Hofweber writes in the Stanford Encyclopedia of Philosophy that logic "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".

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 judgment (German: Urteil). On this conception, the valid inferences of logic follow from the structural features of judgments or thoughts.

History

The earliest sustained work on the subject of logic is that of Aristotle. Aristotelian logic became widely accepted in science and mathematics and remained in wide use in the West until the early 19th century. Aristotle's system of logic was responsible for the introduction of hypothetical syllogism, temporal modal logic, and inductive logic. 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.

The Chinese logical philosopher Gongsun Long (ca. 325–250 BC) proposed the paradox "One and one cannot become two, since neither becomes two." In China, the tradition of scholarly investigation into logic, however, was repressed by the Qin dynasty following the legalist philosophy of Han Feizi.

Logic in Islamic philosophy, particularly Avicenna's logic, was heavily influenced by Aristotelian logic.

In India, innovations in the scholastic school, called Nyaya, continued from ancient times into the early 18th century with the Navya-Nyaya 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. Since 1824, Indian logic attracted the attention of many Western scholars, and has had an influence on important 19th-century logicians such as Charles Babbage, Augustus De Morgan, and George Boole. 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 mid-19th 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 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 logic

Syllogistic 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 logic, also known by the name term logic, are 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 consist of two propositions sharing a common term as premise, and a conclusion which is 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. However, it was not alone: the Stoics proposed a system of propositional logic that was studied by medieval logicians. Also, 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.

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 propositional logic and the predicate calculus. Others use Aristotle in argumentation theory to help develop and critically question argumentation schemes that are used in artificial intelligence and legal arguments.

Propositional logic (aka sentential logic)

A propositional calculus or logic (also a sentential calculus) is a formal system in which formulae representing propositions can be formed by combining atomic propositions using logical connectives, and in which a system of formal proof rules allows certain formulæ to be established as "theorems".

Predicate logic

Predicate logic is the generic term for symbolic formal systems such as first-order logic, second-order logic, many-sorted logic, and infinitary logic.

Predicate logic provides an account of quantifiers general enough to express a wide set of arguments occurring in natural language. Aristotelian syllogistic logic specifies a small number of forms that the relevant part of the involved judgements may take. Predicate logic allows sentences to be analysed into subject and argument in several additional ways, thus allowing predicate logic to solve the problem of multiple generality that had perplexed medieval logicians.

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 Mathematical Logic by David Hilbert and Wilhelm Ackermann in 1928. The analytical generality of predicate logic allowed the formalisation of mathematics, drove the investigation of set theory, and allowed the development of Alfred Tarski's approach to model theory. It provides the foundation of modern mathematical logic.

Frege's original system of predicate logic was second-order, rather than first-order. Second-order logic is most prominently defended (against the criticism of Willard Van Orman Quine and others) by George Boolos and Stewart Shapiro.

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. 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 axiomatizations of the alethic modalities. His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to include deontic logic and epistemic logic. The seminal work of Arthur Prior applied the same formal language to treat temporal logic and paved the way for the marriage of the two subjects. Saul Kripke discovered (contemporaneously with rivals) his theory of frame semantics which revolutionised the formal technology available to modal logicians and gave a new graph-theoretic way of looking at modality that has driven many applications in computational linguistics and computer science, such as dynamic logic.

Informal reasoning

The motivation for the study of logic in ancient times was clear: it is so that one 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. Half of the works of Aristotle's Organon treat inference as it occurs in an informal setting, side by side with the development of the syllogistic, and in the Aristotelian school, these informal works on logic were seen as complementary to Aristotle's treatment of rhetoric.

This ancient 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.

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.

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 mathematics and geometry in relation to logic and philosophy goes back to the ancient Greeks such as Euclid, Plato, and Aristotle. Many other ancient and medieval philosophers applied mathematical ideas and methods to their philosophical claims.

One of the boldest attempts to apply logic to mathematics was undoubtedly the logicism pioneered by philosopher-logicians such as Gottlob Frege and Bertrand Russell: the idea was that mathematical theories were logical tautologies, and the programme was to show this by means to a reduction of mathematics to logic. The various attempts to carry this out met with a series of failures, from the crippling of Frege's project in his Grundgesetze by Russell's paradox, to the defeat of Hilbert's program by Gödel's incompleteness theorems.

Both the statement of Hilbert's program and its refutation by Gödel depended upon their work establishing the second area of mathematical logic, the application of mathematics to logic in the form of proof theory. Despite the negative nature of the incompleteness theorems, Gödel's completeness theorem, a result in model theory and another application of mathematics to logic, can be understood as showing how close logicism came to being true: every rigorously defined mathematical theory can be exactly captured by a first-order logical theory; Frege's proof calculus is enough to describe the whole of mathematics, though not equivalent to it. Thus we see how complementary the two areas of mathematical logic have been.

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 Church-Turing thesis. Today recursion theory is mostly concerned with the more refined problem of complexity classes — when is a problem efficiently solvable? — and the classification of degrees of unsolvability.

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 the invention of mathematical logic. Philosophical logic has a much greater concern with the connection between natural language and logic. As a result, philosophical logicians have contributed a great deal to the development of non-standard logics (e.g., free logics, tense logics) as well as various extensions of classical logic (e.g., modal logics), and non-standard semantics for such logics (e.g., Kripke's technique of supervaluations in the semantics of logic).

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 his 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 formulate an argument correctly.

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.

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 artificial 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, relating to operatives AND, NOT, and OR;

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 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.

Controversies

Just as 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

The logics discussed above are all "bivalent" or "two-valued"; that is, they are most naturally understood as dividing propositions into true and false propositions. Non-classical logics are those systems which reject bivalence.

Hegel developed his own dialectic logic that extended Kant's transcendental logic but also brought it back to ground by assuring us that "neither in heaven nor in earth, neither in the world of mind nor of nature, is there anywhere such an abstract 'either–or' as the understanding maintains. Whatever exists is concrete, with difference and opposition in itself".

In 1910 Nicolai A. Vasiliev rejected the law of excluded middle and the law of contradiction and proposed the law of excluded fourth and logic tolerant to contradiction. In the early 20th century Jan Łukasiewicz investigated the extension of the traditional true/false values to include a third value, "possible", so inventing ternary logic, the first multi-valued logic.

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.

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 non-classical logic. However, modal logic is normally formalised with the principle of the excluded middle, and its relational semantics is bivalent, so this inclusion is disputable.

"Is logic empirical?"

What is the epistemological status of the laws of logic? What sort of argument is appropriate for criticizing 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 anti-realism.

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 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 impossible

Hegel was deeply critical of any simplified notion of the Law of Non-Contradiction. It was based on Leibniz's idea that this law of logic also requires a sufficient ground to specify from what point of view (or time) one says that something cannot contradict itself. A building, for example, both moves and does not move; the ground for the first is our solar system for the second the earth. In Hegelian dialectic, the law of non-contradiction, of identity, itself relies upon difference and so is not independently assertable.

Closely related to questions arising from the paradoxes of implication comes the suggestion that logic ought to tolerate inconsistency. Relevance logic and paraconsistent logic are the most important approaches here, though the concerns are different: a key consequence of classical logic and some of its rivals, such as intuitionistic logic, is that they respect the principle of explosion, which means that the logic collapses if it is capable of deriving a contradiction. Graham Priest, the main proponent of dialetheism, has argued for paraconsistency on the grounds that there are in fact, true contradictions.

Rejection of logical truth

The philosophical vein of various kinds of skepticism contains many kinds of doubt and rejection of the various bases upon which logic rests, such as the idea of logical form, correct inference, or meaning, typically leading to the conclusion that there are no logical truths. Observe that this is opposite to the usual views in philosophical skepticism, where logic directs skeptical enquiry to doubt received wisdoms, as in the work of Sextus Empiricus.

Friedrich Nietzsche provides a strong example of the rejection of the usual basis of logic: his radical rejection of idealisation led him to reject truth as a "mobile army of metaphors, metonyms, and anthropomorphisms—in short ... metaphors which are worn out and without sensuous power; coins which have lost their pictures and now matter only as metal, no longer as coins". His rejection of truth did not lead him to reject the idea of either inference or logic completely, but rather suggested that "logic [came] into existence in man's head [out] of illogic, whose realm originally must have been immense. Innumerable beings who made inferences in a way different from ours perished". Thus there is the idea that logical inference has a use as a tool for human survival, but that its existence does not support the existence of truth, nor does it have a reality beyond the instrumental: "Logic, too, also rests on assumptions that do not correspond to anything in the real world".

This position held by Nietzsche however, has come under extreme scrutiny for several reasons. He fails to demonstrate the validity of his claims and merely asserts them rhetorically. Furthermore, his position has been claimed to be self-refuting by philosophers such as Jurgen Habermas who have accused Nietszche of not even having a coherent perspective let alone a theory of knowledge. George Lukacs in his book The Destruction of Reason has asserted that "Were we to study Nietzsche’s statements in this area from a logico-philosophical angle, we would be confronted by a dizzy chaos of the most lurid assertions, arbitrary and violently incompatible" . Extreme skepticism such as that displayed by nihilists like Nietzsche have not been met with much seriousness by analytic philosophers in the 20th century. Bertrand Russell famously referred to Nietzsche's claims as "empty words" in his book A History of Western Philosophy

See also

Digital electronics (also known as digital logic or logic gates)

Fallacies

Logic puzzle

Logic symbols

Mathematics

* List of mathematics articles

* Outline of mathematics

Metalogic

Outline of logic

Philosophy

* List of philosophy topics

* Outline of philosophy

Reason

Straight and Crooked Thinking (book)

Table of logic symbols

Truth

Notes

References

Nuel Belnap, (1977). A useful four-valued logic. In Dunn & Eppstein, Modern uses of multiple-valued logic. Reidel: Boston.

Cohen, R.S, and Wartofsky, M.W. (1974). Logical and Epistemological Studies in Contemporary Physics. Boston Studies in the Philosophy of Science. D. Reidel Publishing Company: Dordrecht, Netherlands. ISBN 90-277-0377-9.

Finkelstein, D. (1969). "Matter, Space, and Logic". in R.S. Cohen and M.W. Wartofsky (eds. 1974).

Gabbay, D.M., and Guenthner, F. (eds., 2001–2005). Handbook of Philosophical Logic. 13 vols., 2nd edition. Kluwer Publishers: Dordrecht.

Hilbert, D., and Ackermann, W, (1928). Grundzüge der theoretischen Logik (Principles of Mathematical Logic). Springer-Verlag. 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 First-Order Logic. Hackett Publishing.

Kneale, William, and Kneale, Martha, (1962). The Development of Logic. Oxford University Press, London, UK.

Mendelson, Elliott, (1964). Introduction to Mathematical Logic. Wadsworth & Brooks/Cole Advanced Books & Software: Monterey, Calif. OCLC 13580200

Smith, B., (1989). "Logic and the Sachverhalt". The Monist 72(1):52–69.

Whitehead, Alfred North and Bertrand Russell, (1910). Principia Mathematica. Cambridge University Press: Cambridge, England. OCLC 1041146

External links and further readings

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 Self-Taught: A Workbook (originally prepared for on-line logic instruction).

**Nicolas Rescher. (1964). Introduction to Logic, St. Martin's Press.

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. An Introduction with an 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:

**Logic & Metaphysics

**Set Theory and Further Logic

**Mathematical Logic

Category:Philosophy of mathematics Category:Branches of philosophy Category:Abstraction Category:Interdisciplinary fields Category:Formal sciences Category:Greek loanwords Category:Axiology

Name	Donald Fagen
Background	solo_singer
Birth name	Donald Jay Fagen
Born	January 10, 1948
Origin	Passaic, New Jersey
Genre	Rock, jazz fusion
Years active	1971–present
Instrument	piano, organ, various keyboard instruments, melodica, vocals, saxophone
Associated acts	Jay & the Americans, Steely Dan
Label	ABC Records MCA Records Giant/Reprise/Warner Bros. Records Reprise/Warner Bros. Records
Url	Official Site

Donald Jay Fagen (born January 10, 1948) is an American musician and songwriter, best known as the co-founder, lead singer, and the principal songwriter of the rock band Steely Dan (along with partner Walter Becker). Fagen is known for his use of jazz harmonies, elaborate arrangements, and attention to detail. Fagen launched a successful, if sporadic, solo career in 1982, spawning three albums to date.

Early life

Fagen was born in Passaic, New Jersey, on January 10, 1948 to Joseph "Jerry" Fagen, an accountant and his wife Elinor. From the age of 12 to 17, Elinor sang in a hotel band in upstate New York's Catskill Mountains until bouts of stage fright (a condition her son would later be afflicted with) forced her to discontinue performing live. The transition greatly upset Donald: he detested living in the suburbs. He would later recall to an interviewer that it "was like a prison. I think I lost faith in [my parents'] judgement… It was probably the first time I realised I had my own view of life." His life in Kendall Park would later inspire tracks on his album The Nightfly. Fagen would regularly take the bus to Manhattan to see Charles Mingus, Sonny Rollins, Thelonious Monk, and Miles Davis perform.

The family was of the Jewish faith.

After graduating from South Brunswick High School in 1965, Fagen enrolled at Bard College to study English literature, having been inspired by Jack Kerouac, William S. Burroughs, Gregory Corso, Allen Ginsberg, and Lawrence Ferlinghetti. While at Bard, Fagen met musician Walter Becker. The duo, along with a revolving assortment of musicians which included future actor Chevy Chase, formed various groups called The Leather Canary, The Don Fagen Jazz Trio, and the Bad Rock Band. Fagen would later describe his college bands as sounding like "The Kingsmen performing Frank Zappa material." None of the groups lasted long, but the partnership between Fagen and Becker would continue for decades. The duo's early career included a stint with Jay and the Americans under pseudonyms, and in the early 1970s, as pop songwriters, prior to forming Steely Dan.

Career

Steely Dan

Responding to an ad in The Village Voice, Fagen and Becker met Denny Dias and together founded Steely Dan in August 1972. They initially formed the core of the band and co-wrote all the group's music; on tour and record, Becker played bass (and later lead guitar) and Fagen played keyboards, as well as performing almost all of the lead vocals on their recordings.

After releasing their third LP in 1974, the other members gradually left (or were fired from) the band, which gradually evolved into a studio project headed by Becker and Fagen. From the mid-70s onwards, Steely Dan recorded their albums exclusively with jazz and rock session musicians. They scored their biggest success in 1977 with the platinum-selling album Aja.

After a lengthy period of inactivity as a band which began in the early 1980s, the duo of Becker and Fagen revived Steely Dan in the mid 1990s, and have since produced two more Steely Dan studio albums: 2000s Two Against Nature which won several Grammys, Everything Must Go (2003), as well as the live CD Alive in America (1995) and a live concert DVD entitled Two Against Nature, which included material spanning much of the band's history.

Solo career

After Steely Dan's breakup in 1981, Fagen released his critically acclaimed solo debut album, The Nightfly, in October 1982. It reached #11 on the Billboard album chart and was certified Platinum for sales of over a million copies in the United States alone. Its premiere single was "I.G.Y. (What a Beautiful World)". The song hit the Adult Contemporary Top 10 and peaked at #26 on the Billboard Hot 100. The follow-up single, "New Frontier" peaked at #34 AC and #70 Pop and was aided by a popular MTV music video. The Nightfly was nominated for several Grammy Awards including Album Of The Year. It was significantly more jazz-based than Fagen's Steely Dan work. Rhino Records released a special DVD Audio version of The Nightfly in honor of the album's 20th anniversary in 2002.

Fagen's second solo album, 1993's Kamakiriad, was produced by Becker. It climbed into Billboard's Top 10 but sold fewer copies than The Nightfly, topping off at 900,000 in sales. Kamakiriad was also nominated for a Grammy Award for Album Of The Year.

Recording sessions for Fagen's third solo album, Morph the Cat, began in August 2004 and the album was released March 14, 2006. Performing on the album are Wayne Krantz (guitar), Jon Herington (guitar), Keith Carlock (drums), Freddy Washington (bass), Ted Baker (piano), and Walt Weiskopf (sax). Upon its release, Morph the Cat received universal acclaim and was later named Album of the Year by Mix magazine. Fagen was also given 'Producer of the Year' award. The 5.1 mix of Morph the Cat won the 2007 Grammy Award for Best Surround Sound Album.

All three of Fagen's albums previously released on the DVDA format have now been released in Warner's latest format MVI (Music Video Interactive) as a boxed set. Each album features a DTS 5.1, Dolby 5.1 and PCM Stereo mix but no MLP encoded track, along with bonus audio and video content. Customers will also be able to select any track or any portion of a track and use it as their mobile phone ringtone.

In March 2006, Fagen embarked on his first-ever solo tour to support Morph the Cat. This theater tour of the Northeast (including one Canadian date in Toronto), the Midwest, and the West Coast played to sold-out houses and excellent reviews. Fagen's management also sold VIP packages for the tour, which included a pre-show party, premium seats, and access to the band's sound checks. This was mocked on his song-writing partner Walter Becker's website. Fagen joined Becker for a Steely Dan/Michael McDonald tour the following summer of 2007.

He also toured with the New York Rock and Soul Revue. The 1986 debut album by Rosie Vela, Zazu, inspired the first collaboration between Fagen and Becker following the disbanding of Steely Dan. This led to their reunion as a writing team and the creation of a new touring version of Steely Dan. The following year Fagen co-produced Becker's solo debut, 11 Tracks of Whack.

Fagen has stated in a recent interview that he plans to record his fourth solo album in spring of 2010 that will be lighter in tone and not an extension of his Nightfly/Kamikiriad/Morph The Cat trilogy.

Work for, and with, other artists

In 1977, he played synthesizer on Poco's Indian Summer album.

In 1983, he played synthesizer for the track "Love Will Make It Right," which he also wrote, on Diana Ross's solo album Ross.

In 1986, he wrote the title track to the Yellowjackets' album Shades.

Also in 1986, Fagen and Becker contributed to Rosie Vela's album Zazu

In 1991, The Manhattan Transfer recorded Fagen's song "Confide in Me" on their album The Offbeat of Avenues.

Fagen's own versions of both "Confide in Me" and "Big Noise New York" were released as 'B' sides for the CD singles of the Kamakiriad tracks "Tomorrow's Girls" and "Snowbound," respectively. They were subsequently made available on the Bonus CD of The Nightfly Trilogy boxed set in 2007.

In 2008, Fagen played synthesizer on the album I Know You're Married But I've Got Feelings Too by folk singer Martha Wainwright.

Since 2008, Fagen has made regular appearances with the Levon Helm Band, performing at Levon's Midnight Ramble concerts in Woodstock, New York. He plays keyboards and sings with his step-daughter, Amy Helm (Amy's mother is Fagen's wife, Libby Titus).

Soundtrack contributions

He recorded "True Companion" for the 1981 animated cult film Heavy Metal.

He wrote "The Finer Things", which appeared on the soundtrack to the Martin Scorsese 1983 film The King of Comedy, as well as sang background vocals on the song. David Sanborn is given credit for the main part of the song.

He recorded "Century's End" for 1988's Bright Lights, Big City.

He recorded "Reflections", a Thelonious Monk cover, along with guitarist Steve Khan, for the film 1988 .

Other works

Fagen wrote briefly in the 1980s for Premiere magazine, including a few witty pieces on Henry Mancini and Ennio Morricone. These are available online from his website. His website also features other articles he has written, such as ones for Slate.com.

In 1988, Fagen was a co-producer of the Broadway soundtrack album The Gospel at Colonus.

In the late 2000s, he appeared as the "Wise Man" in a promotional video for the Electro-Harmonix Stereo Memory Man with Hazarai digital delay/looper. This short promo video was directed by frequent Steely Dan trumpet player Michael Leonhart.

Discography

1982 The Nightfly – #11 US; #44 UK

1993 Kamakiriad – #10 US; #3 UK

2006 Morph the Cat – #26 US; #35 UK

2007 The Nightfly Trilogy (3-MVI DVD + 4-CD Box Set)

2011 untitled fourth solo album

Awards

In 1984, Fagen was awarded an Honorary Doctor of Arts degree by his alma mater, Bard College.

In 2001, both Fagen and Becker received Honorary Doctor of Music degrees from Berklee College of Music. They both accepted their degrees in person.

In 2001, Steely Dan (Fagen and Becker) were inducted into the Rock and Roll Hall of Fame.

Personal life

In 1993, Fagen married fellow songwriter Libby Titus. Although the two of them had attended Bard College at around the same time, they didn't become friends until 1987 when they were both going backstage after a Dr. John concert. Libby Titus co-wrote the song "Florida Room", which appears on 1993's album Kamakiriad.

Quotation

NME - January 1983

References

External links

donaldfagen.com

Category:Steely Dan members Category:American male singers Category:American rock musicians Category:American rock pianists Category:Songwriters from New Jersey Category:Bard College alumni Category:Jewish American musicians Category:People from Passaic County, New Jersey Category:1948 births Category:Living people Category:Grammy Award winners Category:Jewish composers and songwriters Category:People from Middlesex County, New Jersey

Name	Christian Louboutin
Birth date	January 07, 1964
Birth place	France
Occupation	Shoe designer
Years active	1991–present

Christian Louboutin (pronounced )) (born January 7, 1964) is a footwear designer who launched his line of high-end women's shoes in France in 1991. Since 1992, his designs have incorporated the shiny, red-lacquered soles that have become his signature. On March 27, 2007, Louboutin filed an application for U.S. trademark protection of this red sole design.

Biography

As a child, Louboutin would regularly sneak out of school, from the age of 12, to watch the showgirls at Paris nightclubs, because he was fascinated by their costumes. He cites this as his main inspiration for becoming a shoe designer: "[The showgirls] influenced me a lot. If you like high heels, it's really the ultimate high heel - it's all about the legs, how they carry themselves, the embellishment of the body. They are the ultimate icons." Although Louboutin faced much opposition following his decision to leave school so early, he claims that his resolve was strengthened after watching an interview on TV with Sophia Loren in which she introduced her sister, saying she had to leave school when she was only 12 but when she turned 50 she got her degree. "Everybody applauded! And I thought, 'Well, at least if I regret it I'm going to be like the sister of Sophia Loren!"

Shoes

Louboutin helped bring stilettos back into fashion in the 1990s and 2000s, designing dozens of styles with heel heights of 120mm (4.72 inches) and higher. The designer's professed goal is to “make a woman look sexy, beautiful, to make her legs look as long as [he] can.” While he does offer some lower-heeled styles, Louboutin is generally associated with his dressier eveningwear designs incorporating bejeweled straps, bows, feathers, patent leather and other, similar decorative touches.

In his U.S. trademark application, Louboutin explains the inception of the signature red soles:

"In 1992 I incorporated the red sole into the design of my shoes. This happened by accident as I felt that the shoes lacked energy so I applied red nail polish to the sole of a shoe. This was such a success that it became a permanent fixture."

Louboutin received inspiration for his lethal-looking stilettos from an incident that occurred in his early 20s. He had visited a museum and noticed that there was a sign forbidding women wearing sharp stilettos from entering for fear of damage to the extensive wood flooring. This image stayed in his mind, and he later used this idea in his designs. "I wanted to defy that," Louboutin has said. "I wanted to create something that broke rules and made women feel confident and empowered."

Louboutin has topped The Luxury Institute's annual Luxury Brand Status Index (LBSI) for three years; the brand's offerings were declared the Most Prestigious Women's Shoes in 2007, 2008, and 2009.

Style Gallery: Selected Christian Louboutin Styles

Stores

There are three Christian Louboutin boutiques in Paris, located on Rue Jean-Jacques Rousseau, Rue du Faubourg St-Honoré and Rue de Grenelle.

Christian Louboutin has seven boutiques in the United States: two in New York (Madison & Horatio), two in California (West Hollywood & South Coast Plaza), one in Las Vegas, one in the Design District in Miami and is opening one in Highland Park Village in Dallas.

There are two in London, England. The original is in Motcomb Street. An additional one opened in Mount Street in 2008. There are Christian Louboutin concessions at Harvey Nichols, Harrods and Selfridges.

In Asia, the first Christian Louboutin boutique was opened in October 2007 on On Lan Street in Central, Hong Kong. Other boutiques in Asia are located in Singapore's Ngee Ann City mall as well as in Jakarta, Indonesia.

In Australia, you can find his shoes sold in Melbourne, Brisbane, Sydney, Adelaide and Perth in David Jones stores.

The first Christian Louboutin store in Australia is in Sydney's Westfield Sydney

Recently, Louboutin opened his first store in South America in Shopping Iguatemi São Paulo.

Fighting online fraud

Branded genuine Louboutin shoes are sold through various luxury goods retailers such as Harrods, Harvey Nichols, Selfridges and net-a-porter in the UK and Barneys, Saks Fifth Avenue, Neiman Marcus, Bergdorf Goodman, and Nordstrom in the US. Online, the Louboutin brand is under constant attack from fake and knock-off cheaply made versions, often made in China. Louboutin's own website contains a prominent note stating that they do not sell their products online; and that any other domain name containing the word "Louboutin" is very likely to be selling counterfeit goods.

In the last few years, the company has served hundreds of DMCA notices on Google to remove many sites selling fake goods from their search results. Even after this action, thousands more sites remain online.

The company has recently set up a separate website focused on protecting their brand, and drawing attention to the problems. It details hundreds of websites selling fake goods, as well as summaries of legal action taken including raids on factories and photographs and videos of the mass destruction of counterfeit goods discovered.

Media

Jennifer Lopez recorded a song about the shoes, naming it "Louboutins". It will be released as the first single from her album Love?.

The Oprah Winfrey Show featured Louboutin and a collection of his shoes in a show dedicated to "the face behind the name"; Louboutins were on display around the set, with Oprah Winfrey describing the shoes as "pieces of art".

Louboutin has recently collaborated with director David Lynch and they have created an exhibition featuring rare shoes by Louboutin, they are on show at the Pierre Passebon Gallery. The exhibition is named "Fetish".

A popular Persian rap band, ZedBazi, mentions Christian Louboutin in their latest song Messe Man.

According to a reviewer, Louboutin shoes were worn by rising television morning-news-show producer Becky Fuller (Rachel McAdams) in a scene as she plays the title character in the 2010 movie Morning Glory.

Louboutins play a key role in the 2010 Motion Picture Burlesque, starring Christina Aguilera and Cher. Aguilera's character spots a pair of Louboutins at a party and a male interest ultimately purchases the shoe for her.

Britney Spears is an avid Louboutin fan.

Britney Spears in Louboutin for a photoshoot in 2009.

References

External links

Christian Louboutin Official Site (in English)

Category:Shoe designers Category:French fashion designers Category:1964 births Category:Living people Category:Luxury brands