Logic (from the Greek λογική logikē)^[1] is the philosophical study of valid reasoning.^[2] Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science. It examines general forms that arguments may take, which forms are valid, and which are fallacies. In philosophy, the study of logic is applied in most major areas: metaphysics, ontology, epistemology, and ethics. In mathematics, it is the study of valid inferences within some formal language.^[3] Logic is also studied in argumentation theory.^[4]

Logic was studied in several ancient civilizations, including India,^[5] China,^[6] and Greece. In the West, logic was established as a formal discipline by Aristotle, who gave it a fundamental place in philosophy. The study of logic was part of the classical trivium, which also included grammar and rhetoric.

Logic is often divided into three parts, inductive reasoning, abductive reasoning, and deductive reasoning.

The study of logic[link]

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^[7] 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.^[8] 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.^[9][10] 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[link]

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 into 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'.^[11] 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 Failed to parse (Missing texvc executable; please see math/README to configure.): m(x)

and Failed to parse (Missing texvc executable; please see math/README to configure.): d(x)

, and the sentence is given by Failed to parse (Missing texvc executable; please see math/README to configure.): \forall x. (m(x) \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, and retroductive inference[link]

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. Similarly, it is important to distinguish 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.

Retroductive inference is a mode of reasoning that Peirce proposed as operating over and above induction and deduction to “open up new ground” in processes of theorizing (1911, p. 2). He defines retroduction as a logical inference that allows us to "render comprehensible" some observations/events which we perceive, by relating these back to a posited state of affairs that would help to shed light on the observations (Peirce, 1911, p. 2). He remarks that the “characteristic formula” of reasoning that he calls retroduction is that it involves reasoning from a consequent (any observed/experienced phenomena that confront us) to an antecedent (that is, a posited state of things that helps us to render comprehensible the observed phenomena). Or, as he otherwise puts it, it can be considered as “regressing from a consequent to a hypothetical antecedent” (1911, p. 4). See for instance, the discussion at: http://www.helsinki.fi/science/commens/dictionary.html

Some authors have suggested that this mode of inference can be used within social theorizing to postulate social structures/mechanisms that explain the way that social outcomes arise in social life and that in turn also indicate that these structures/mechanisms are alterable with sufficient social will (and visioning of alternatives). In other words, this logic is specifically liberative in that it can be used to point to transformative potential in our way of organizing our social existence by our re-examining/exploring the deep structures that generate outcomes (and life chances for people). In her book on New Racism (2010) Norma Romm offers an account of various interpretations of what can be said to be involved in retroduction as a form of inference and how this can also be seen to be linked to a style of theorizing (and caring) where processes of knowing (which she sees as dialogically rooted) are linked to social justice projects (http://www.springer.com/978-90-481-8727-0)

Consistency, validity, soundness, and completeness[link]

Among the important properties that logical systems can have:

• Consistency, which means that no theorem of the system contradicts another.^[12]
• 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).^[12]
• Completeness, of a logical system, which means that if a formula is true, it can be proven (if it is true, it is a theorem of the system).
• Soundness, the term soundness has multiple separate meanings, which creates a bit of confusion throughout the literature. Most commonly, soundness refers to logical systems, which means that if some formula can be proven in a system, then it is true in the relevant model/structure (if A is a theorem, it is true). This is the converse of completeness. A distinct, peripheral use of soundness refers to arguments, which means that the premises of a valid argument are true in the actual world.

Some logical systems do not have all four properties. As an example, Kurt Gödel's incompleteness theorems show that sufficiently complex formal systems of arithmetic cannot be consistent and complete;^[10] however, first-order predicate logics not extended by specific axioms to be arithmetic formal systems with equality can be complete and consistent.^[13]

Rival conceptions of logic[link]

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".^[3]

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[link]

Aristotle, 384–322 BC.

The earliest sustained work on the subject of logic is that of Aristotle.^[14] Aristotelian logic became widely accepted in science and mathematics and remained in wide use in the West until the early 19th century.^[15] Aristotle's system of logic was responsible for the introduction of hypothetical syllogism,^[16] temporal modal logic,^[17][18] and inductive logic.^[19] In Europe during the later medieval period, major efforts were made to show that Aristotle's ideas were compatible with Christian faith. During the High 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."^[20] In China, the tradition of scholarly investigation into logic, however, was repressed by the Qin dynasty following the legalist philosophy of Han Feizi.

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.^[21] 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.^[22] 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^[9] 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[link]

Syllogistic logic[link]

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.^[23] 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 (sentential logic)[link]

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 formulae to be established as "theorems".

Predicate logic[link]

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[link]

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,^[24] who was concerned with the alethic modalities of necessity and possibility, which he observed to be dual in the sense of De Morgan duality.^{[citation needed]} 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[link]

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[link]

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.^[25]

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.^[26] Many other ancient and medieval philosophers applied mathematical ideas and methods to their philosophical claims.^[27]

Kurt Gödel

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.^[9] 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.^[28] Despite the negative nature of the incompleteness theorems, Gödel's completeness theorem, a result in model theory and another application of mathematics to logic, can be understood as showing how close logicism came to being true: every rigorously defined mathematical theory can be exactly captured by a first-order logical theory; Frege's proof calculus is enough to describe the whole of mathematics, though not equivalent to it. Thus we see how complementary the two areas of mathematical logic have been.^{[citation needed]}

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.^[29] 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.^[30]

Philosophical logic[link]

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.

Computational Logic[link]

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.

Bivalence and the law of the excluded middle[link]

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".^[31]

In 1910 Nicolai A. Vasiliev extended the law of excluded middle and the law of contradiction and proposed the law of excluded fourth and logic tolerant to contradiction.^[32] 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.^{[citation needed]}

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.^[33]

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 can be applied for extracting verified programs from proofs.

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?"[link]

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?"^[34] 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.^[35]

Another paper by the same name by Sir Michael Dummett argues that Putnam's desire for realism mandates the law of distributivity.^[36] 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?[link]

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 since granny is mortal, 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[link]

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 and 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.^[37]

Rejection of logical truth[link]

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".^[38] 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".^[39] 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".^[40]

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 Jürgen Habermas, who have accused Nietszche of not even having a coherent perspective let alone a theory of knowledge.^[41] 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".^[42] Extreme skepticism such as that displayed by Nietzsche has 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.^[43]

See also[link]

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

Logic portal List of logic journals Philosophy List of philosophy topics Outline of philosophy Reason Straight and Crooked Thinking (book) Table of logic symbols Truth

Notes[link]

References[link]

External links and further readings[link]

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	Book: Logic
Wikipedia books are collections of articles that can be downloaded or ordered in print.

• LogicWiki, an external 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 Self-Taught: A Workbook (originally prepared for on-line logic instruction).
    • Nicholas 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

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Occupations	Musician, songwriter
Instruments	Vocals, keyboards, guitar, bass
Years active	1969–present
Labels	A&M, Unichord/Voiceprint, Epic
Associated acts	Supertramp, Ringo Starr and His All-Starr Band, Argosy
Website	RogerHodgson.com

Charles Roger Pomfret Hodgson (born 21 March 1950) is a British musician and songwriter, best known as the former co-frontman, and founding member, of progressive rock band Supertramp.^[1][2][3]

Hodgson composed and sang on the majority of the band's hits, including "Dreamer", "Give a Little Bit", "The Logical Song", "Breakfast in America", "Take the Long Way Home" and "It's Raining Again".

He left Supertramp in 1983 to pursue a solo career, but after two albums he retired from touring and other industry obligations. In 1997 he returned to doing solo tours, and released a third album in 2000. He is recognised for his tenor singing voice, which became a trademark of his former band Supertramp, and often writes about spiritual and philosophical topics.

Biography[link]

1950–1969: Early years[link]

Hodgson was born in Portsmouth, Hampshire, England, in 1950 and grew up in Oxford. He is the son of Charles Hodgson and Jill Hodgson (died in June 2009). He attended Woodcote House near Wallingford, Oxfordshire, where he was the first boy to learn electric guitar, and Stowe School near Buckingham, Buckinghamshire. His first guitar was a parting gift from his father at age 12 when his parents divorced.^[4] He took it to boarding school with him, where his teacher taught him three chords. He began composing his own music and lyrics and within a year gave his first concert at school with nine original songs. Hodgson's first band at school consisted of him on guitar and his friend Roy Hoby playing snare drums. They were dubbed the "H-bombs" because of their last names.

At age 19, Roger Hodgson made his first appearance in a recording studio as guitarist for People Like Us, a band he formed shortly after graduating from boarding school.^[4] The group recorded a single, "Duck Pond" b/w "Send Me No Flowers", which was never released.^[5]

After People Like Us disbanded, Hodgson auditioned for Island Records, with Traffic's road manager providing him a foot in the door with the label.^[4] Island set him up in a recording studio as vocalist for the one-off "flower power" pop band Argosy, which also included Reginald Dwight (later known as Elton John), Caleb Quaye, and Nigel Olsson.^[6] Their sole single, "Mr. Boyd" b/w "Imagine", consisted of two pieces of orchestrated pop (both penned by Hodgson) and was issued in 1969 on the DJM (U.K.) and Congress (U.S.) record labels.^[6] It sold poorly and consequently has become rare and sought after. "Mr. Boyd" was covered in 1997 by Jake Shillingford and his band My Life Story on their album "The Golden Mile".

1969–1983: Supertramp[link]

After the breakup of Argosy, Hodgson, responding to an advert placed in Melody Maker by Rick Davies, auditioned for the guitarist spot in the progressive rock band Supertramp. Hodgson was offered the job, but when Richard Palmer arrived the next day to audition for the same spot, Hodgson agreed to learn how to play bass instead.^[7]

The songs on Supertramp's self-titled first album, released in 1970, were composed by Roger Hodgson, Rick Davies, and Richard Palmer; however, since both Hodgson and Davies were unwilling to write lyrics, Palmer wrote all the album's lyrics.^[7] Palmer left shortly after the album's recording, allowing Hodgson to switch back to guitar, but leaving him and Davies no choice but to serve as the band's lyricists. Paralleling fellow British prog rockers Genesis and their search for a new lead vocalist, Supertramp auditioned 93 guitarists before turning the role over to Hodgson.^[8] The hugely successful Crime of the Century was released in 1974. Crisis? What Crisis?, released in 1975, was followed by Even in the Quietest Moments in 1977. In 1979, they released their most successful album, Breakfast in America. This album has sold over 20 million copies to date.^[9] The live album, Paris, was released in 1980. …Famous Last Words…, released in 1982, included Hodgson's first solo recording, "Know Who You Are".

From 1974 through 1983, all songs recorded by Supertramp were legally credited with a shared writing credit of Davies/Hodgson. Roger Hodgson was the writer of hits such as "Give a Little Bit", "Breakfast in America", "It's Raining Again", "Take the Long Way Home" and "Fool's Overture". Hodgson wrote "Breakfast in America", "The Logical Song", and some of "Fool's Overture" at home with a harmonium he had bought from a neighbour when he was 17 years old (this instrument is used in the background of "Breakfast in America", and prominently appears on "Two of Us" and his solo track "The Garden").

In 1981, Hodgson left Los Angeles and moved his family to Northern California, where he built a home studio and began contemplating solo recordings.^[10] The rest of Supertramp remained in Los Angeles, and the geographic separation created a rift between them and Hodgson; there was little to no feuding, but the group harmony was lost. Hodgson felt increasingly constrained in the group context, and during the tour for …Famous Last Words… he made the final decision to leave Supertramp.^[11] He has stated that there were not any real problems in his relationship with Davies, as was speculated.^[12]

1984–present: Solo career[link]

Hodgson recorded three solo albums at his new home studio, the first before his departure from Supertramp. Titled Sleeping With the Enemy, it was cut in the months between the release of …Famous Last Words… and its supporting tour, and mixed during Supertramp rehearsals for said tour in hopes of fitting in some solo promotion while on the road.^[11] However, at the last minute Hodgson had second thoughts about the album's quality and decided to scrap it, planning to record a new and better album after his last tour with Supertramp.^[11]

This second effort, In the Eye of The Storm, was released in 1984. Despite being heavily publicized as the solo album of a former member of Supertramp, it failed to break the top 40 in either the US or UK.^[13][14] The single "Had a Dream (Sleeping With the Enemy)" was able to reach no higher than number 48 in the US,^[15] while the follow-up single, "In Jeopardy", failed to chart at all.

Though a major commercial disappointment after his last six albums with Supertramp, In the Eye of the Storm would prove to be Hodgson's biggest success without the group. His second album was Hai Hai (1987), however, just prior to the release of Hai Hai, Hodgson fell from a loft in his home and broke both wrists, which disabled him from promoting the album.^[16] It would barely scrape into the Billboard 200,^[17] and did not make the UK Chart. He decided to take a long break from both touring and recording in order to spend more time with his children.^[16]

In 1990, Hodgson was approached by Yes to take the lead singer position but declined the offer. One of the songs he co-wrote with Trevor Rabin, "Walls", appears on Yes's 1994 Talk album, with lyrics revised by Anderson. A version of "Walls" with only Hodgson and Rabin on vocals was released on Trevor Rabin's 2003 archival release 90124.

After a long break, he launched into his first tour in over ten years, and released 1997's live Rites of Passage to document the tour. The live album was recorded at the Miners Foundry in Nevada City, California. He performed with a full band including his son Andrew and Supertramp sax player John Helliwell. The album was a total flop in both the UK and US, but gave Hodgson his only significant solo success in Germany, reaching number 34 in the charts there.^[18]

Hodgson played King Arthur in the rock opera Excalibur: La Legende Des Celtes, and appeared on the album for two songs: "The Elements," and "The Will of God." The project was headed by Alan Simon and released in 1999. In 2000, he contributed vocals on a track titled "The Moon Says Hello" by Carlos Núñez, on the CD Mayo Longo.

Hodgson's fourth solo effort Open the Door was released in 2000 and continued in the vein of his previous work. He collaborated again with Alan Simon on the album. In August 2000, Hodgson guested with Fairport Convention at that years Cropredy Festival. He performed "Breakfast In America", "The Logical Song", "Open The Door" and "Give A Little Bit".

Hodgson toured as a member of the All-Starr Band in 2001, playing lead guitar, and has since collaborated with Trevor Rabin (who appears on the track "The More I Look" on Open the Door) and Ringo Starr.

Hodgson is still touring, often playing alone, but from time to time he is joined by other musicians or a full orchestra. He took part in the Night of the Proms concert series in Belgium and Germany in late 2004, as well as the rock festival Bospop in 2005. On 30 November 2005, he held his first concert in England in over twenty years, at Shepherd's Bush, London. While the performance was filmed and scheduled for a DVD release, the plan was scrapped. Instead, the concert recorded at the Place Des Arts in Montreal, Canada on 6 June 2006 was his first DVD, released on 22 August 2006, entitled Take The Long Way Home - Live In Montreal. In October 2006, the DVD was certified multi-platinum by the CRIA (Canadian Recording Industry Association).

In May 2006, Hodgson was honoured by ASCAP in recognition of his song "Give A Little Bit" being one of the most played songs in the ASCAP repertoire in 2005. He received another ASCAP award on 9 April 2008 for the Gym Class Heroes' song "Cupid's Chokehold", recognized as one of the most played songs in ASCAP's repertoire in 2007.

Hodgson participated as a mentor on Canadian Idol along with Dennis DeYoung. He continued mentoring several of the finalists during his 2006 Canadian Tour.

Hodgson performed at the Concert for Diana at Wembley Stadium, UK on 1 July 2007. He sang a medley of his most popular songs: "Dreamer", "The Logical Song", "Breakfast in America" and "Give A Little Bit".

On 18 September 2007, Eagle Vision released the DVD Take the Long Way Home - Live in Montreal worldwide, achieving gold in Germany and France.

Hodgson toured the US, Australia, New Zealand, South America, Europe, and Canada in 2010. Though Hodgson's former bandmates in Supertramp announced a 40th Anniversary reunion tour, he was not invited to join them, and his own touring schedule would have prohibited him from any participation beyond the occasional guest spot in any case.^[19] Both Hodgson and Supertramp released tour material on download only on their websites. Hodgson's Classics Live is a collection of recordings taken from solo, band, and orchestra shows from his 2010 world tour. Hodgson again toured worldwide in 2011 and is again on a world tour which started in February and will finish in November.^[20]

Personal life[link]

Hodgson has been a vegetarian all of his life. He said in a 1983 interview about the topic: "I've always loved going out in the woods with all the trees because I also have the respect for the wilderness because I am a vegetarian and I don't eat meat."^{[citation needed]}

Discography[link]

For his work with Supertramp, see Supertramp discography between 1969 and 1983

References[link]

External links[link]

Wikimedia Commons has media related to: Roger Hodgson

• Official website
• Roger Hodgson on Myspace
• UK fansite

v t e Supertramp Rick Davies Bob Siebenberg John Helliwell Carl Verheyen Lee Thornburg Cliff Hugo Jesse Siebenberg Gabe Dixon Cassie Miller-Thornburg Past members: Roger Hodgson Dougie Thomson Richard Palmer Robert Millar Frank Farrell Kevin Currie Dave Winthrop Mark Hart Tom Walsh Keith Baker Touring members: Marty Walsh (1985–1988) Brad Cole (1986–1988) Scott Page (1983–1985) Fred Mandel (1983) Albums Supertramp Indelibly Stamped Crime of the Century Crisis? What Crisis? Even in the Quietest Moments... Breakfast in America ...Famous Last Words... Brother Where You Bound Free as a Bird Some Things Never Change Slow Motion Live albums Concert Classics Paris Live '88 It Was the Best of Times Is Everybody Listening? 70-10 Tour Compilations The Autobiography of Supertramp The Very Best of Supertramp The Very Best of Supertramp 2 Retrospectacle – The Supertramp Anthology Singles "Forever"" "Land Ho" "Dreamer" "Bloody Well Right" "Lady" "Ain't Nobody But Me" "Give a Little Bit" "Babaji" "From Now On" "The Logical Song" "Breakfast in America" "Goodbye Stranger" "Take the Long Way Home" "Dreamer (live)" "Breakfast in America (live)" "It's Raining Again" "Don't Leave Me Now" "Crazy" "My Kind of Lady" "Still in Love" "Cannonball" "Better Days" "I'm Beggin' You" "Free As A Bird" "Give a Little Bit (New Version)" "You Win, I Lose" "Listen To Me, Please" "Slow Motion" Selected songs "Brother Where You Bound" "Fool's Overture" Related articles Discography Band members Book:Supertramp

Maverick Sabre (born Michael Stafford; 12 July 1990 in Hackney, London)^[1] is an Irish-English singer-songwriter and rapper.^[2] Associated with genres such as R&B and folk, Sabre is known for his eerie vocals and guitar playing. In January 2011, Sabre appeared on Professor Green's single "Jungle", which peaked at number 31 in the UK Singles Chart.^[3] His debut EP, The Lost Words was released on 7 March 2011, with the single "Look What I've Done" preceding that release the week before.^[4]

Career[link]

Early history[link]

Maverick Sabre was born in Hackney, London.^[2] At the age of four Sabre moved to Ireland, where he grew up in New Ross, County Wexford, but later - following the suggestion of UK singer/rapper Plan B - returned to London to pursue a musical career.^[2][5][6]

2010: Breakthrough[link]

His career began in 2008 when he appeared on the album Sense The Terror by Irish rappers Terawrizt and Nu-Centz. Sabre appeared on two tracks: "Lucky" and "Used To Blame".^[7] He also featured on rappers Johnnyboy and Intox's 'Propaganda EP' and released his own tracks such as "Lonely Side Of Life" and "Fade Away". Sabre furthered his career in 2009 when he featured on Jermicide & Danny Digg's album Middle Class, collaborating on three tracks: "Afraid", "Where We Go" and "Still With Me".^[8] Maverick Sabre also performed on SB.TV with the song "They Found Him A Gun"^[9] and featured in BBC 1Xtra 'Jail Tales' programme, performing "These Walls".^[10] Sabre then returned to working with his earlier cohorts, featuring on Terawrizt's album Written in Stone and Terawrizt & Nu-Centz' album Sense The Terror 2, appearing on tracks such as "Life Fly By". On 25 November 2010, Sabre released his debut mixtape, The Travelling Man, in the United Kingdom. The release was a free digital download and featured the track "Sometimes" along with a promotional video to accompany the song. On 6 January 2011, "Sometimes" was released as a promotional single without rapper Wretch 32. Sabre guested on drum and bass duo Chase & Status's second studio album No More Idols, on "Fire In Your Eyes". He also features on "Sleepless", a song regularly performed live by Chase & Status, however no studio version has been released to date. Sabre featured on Professor Green’s fourth single "Jungle", released on 1 January 2010, which was featured on the album Alive Till I'm Dead.^[11] The single debuted at number 34 on the UK Singles Chart, peaking at 31 the following week.^[3] On 28 February 2011, Maverick released his debut single "Look What I Done", taken from the extended play The Lost Words, released on 7 March 2011.^[4] On 19 April 2011, Sabre performed with a live band on BBC Two's Later… with Jools Holland.^[12] After his performance of Oasis's "Wonderwall" on Chris Moyles' BBC Radio 1 Comic Relief show, Maverick's extended play The Lost Words reached number 2 in the iTunes Store Top Ten Songs chart. In 2011, he also appeared on The Script's Science & Faith Tour supporting the Irish band, and performed his song "Let Me Go", which entered the UK Singles Chart at number 16 in August 2011.

[edit] 2011-present: Lonely Are the Brave

On 25 July 2011, Maverick Sabre's first single "Let Me Go" was released and entered the UK chart at number 16. He released his debut album, Lonely Are The Brave, on 6 February 2012.^[13] He then released singles "I Need", "What Have I Done To You" and "No One" Maverick Sabre also featured on a remix of his song 'I Need' with rappers Chipmunk and Benny Banks. He then featured on the track 'We'll Never Know' alongside rappers Akala and Logic. On Sunday 29th January 2012, he appeared on the Sunday Night Sessions segment of The Late Show with Joanne Good on BBC London 94.9^[14].

Discography[link]

• Lonely Are the Brave (2012)

References[link]

Jim Jones
Born	James W. Jones (1931-05-13)May 13, 1931 Randolph County, Indiana, U.S.
Died	November 18, 1978(1978-11-18) (aged 47) Jonestown, Guyana
Occupation	Religious leader
Religion	Agnostic/Atheist[1]
Spouse	Marceline Baldwin Jones (1927 - 1978)
Children	Agnes Paulette Jones (1943 - 1978) Suzanne Jones Cartmell (1953 - 2006) Stephanie Jones (1954 - 1959) Lew Eric Jones (1956 - 1978) Jim Jon Prokes (1975 - 1978) Stephan Gandhi Jones (1958- ) James Warren Jones, Jr. (1961-)
Parents	James Thurman Jones (1887 - 1951) Lynetta Putnam Jones (1902 - 1977)

Reverend James Warren "Jim" Jones (May 13, 1931 – November 18, 1978) was the founder and leader of the Peoples Temple, which is best known for the November 18, 1978 mass suicide of 909 Temple members in Jonestown, Guyana along with the killings of five other people at a nearby airstrip. Over 200 children were murdered at Jonestown, almost all of whom were forcibly made to ingest cyanide by the elite Temple members.

Jones was born in Indiana and started the Temple in that state in the 1950s. Jones and the Temple later moved to California, and both gained notoriety with the move of the Temple's headquarters to San Francisco in the mid-1970s.

The incident in Guyana ranks among the largest mass suicides in history, though most likely it involved forced suicide and/or murder, and was the single greatest loss of American civilian life in a non-natural disaster until the events of September 11, 2001. Among the dead was Leo Ryan, who remains the only Congressman assassinated in the line of duty as a Congressman in the history of the United States.^[2]