From Wikipedia, the free encyclopedia
Logic, from the Greek λογικός (logikos)
^{[1]} is the study of
reasoning.
^{[2]} Logic is used in most intellectual activity, 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 falls in the area of
epistemology, which asks: "How do we know what we know?" In mathematics, it is the study of valid
inferences within some
formal language.
^{[3]}
As a discipline, logic dates back to
Aristotle, who established its fundamental place in
philosophy. The study of logic is part of the classical
trivium.
Averroes defined logic as "the tool for distinguishing between the true and the false"
^{[4]};
Richard Whately, '"the Science, as well as the Art, of reasoning"; and
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.
Nature of logic
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^{[6]} 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.^{[7]} 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.^{[8]}^{[9]} 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 analyse 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'.
^{[10]} 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 nonlogical 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 nonlogical concepts, here analysed as m(x) and d(x), and the sentence is given by , involving the logical connectives for universal quantification and implication.
 But equally, the modern view is more powerful: medieval logicians recognised 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 recognise 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 and 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 wellunderstood notions of
semantics. Inductive validity on the other hand requires us to define a reliable generalization of some set of observations. The task of providing this definition may be approached in various ways, some less formal than others; some of these definitions may use
mathematical models of probability. For the most part this discussion of logic deals only with deductive logic.
Consistency, validity, soundness, and completeness
 Consistency, which means that no theorem of the system contradicts another.^{[11]}
 Validity, which means that the system's rules of proof will never allow a false inference from true premises.^{[11]}
 Soundness, which means that the system's rules of proof will never allow a false inference from true premises, and the premises prove true. Soundness results from both validity and true premises. If a system is sound and its axioms are true then its theorems are also guaranteed to be true.^{[11]}
 Completeness, which means that if a theorem is true, it can be proven.
Some logical systems do not have all three properties. As an example,
Kurt Gödel's
incompleteness theorems show that no standard formal system of arithmetic can be consistent and complete.
^{[9]} At the same time his theorems for firstorder predicate logics not extended by specific axioms to be arithmetic formal systems with equality, show those to be complete and consistent.
^{[12]}
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".
^{[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 of logic
The earliest sustained work on the subject of logic is that of
Aristotle,
^{[13]} In contrast with other traditions,
Aristotelian logic became widely accepted in science and mathematics, ultimately giving rise to the formally sophisticated systems of modern logic.
Several ancient civilizations have employed intricate systems of reasoning and asked questions about logic or propounded logical paradoxes. In
India, the
Nasadiya Sukta of the
Rigveda (
RV 10.129) contains
ontological speculation in terms of various logical divisions that were later recast formally as the four circles of
catuṣkoṭi: "
A", "
not A", "
Neither A or not A", and "
Both not A and not not A".
^{[14]} The
Chinese philosopher
Gongsun Long (ca. 325–250 BC) proposed the paradox "One and one cannot become two, since neither becomes two."
^{[15]} Also, the Chinese
School of Names is recorded as having examined logical puzzles such as "A White Horse is not a Horse" as early as the fifth century BCE.
^{[16]} 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, though it did not survive long into the
colonial period. In the 20th century, Western philosophers like Stanislaw Schayer and Klaus Glashoff have tried to explore certain aspects of the
Indian tradition of logic.
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.
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, were the analysis of the judgements into propositions consisting of two terms that are related by one of a fixed number of relations, and the expression of inferences by means of
syllogisms that consisted of two propositions sharing a common term as premise, and a conclusion which was a proposition involving the two unrelated terms from the premises.
Aristotle's work was regarded in classical times and from medieval times in Europe and the Middle East as the very picture of a fully worked out system. It was not alone: the
Stoics proposed a system of
propositional logic that was studied by medieval logicians; nor was the perfection of Aristotle's system undisputed; for example the
problem of multiple generality was recognised in medieval times. Nonetheless, problems with syllogistic logic were not seen as being in need of revolutionary solutions.
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.
Sentential (propositional) 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 a system of formal proof rules allows certain formulæ to be established as "theorems".
Predicate 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.
Modal logic
Main article:
Modal logic
In languages,
modality deals with the phenomenon that subparts of a sentence may have their semantics modified by special verbs or modal particles. For example, "
We go to the games" can be modified to give "
We should go to the games", and "
We can go to the games"" and perhaps "
We will go to the games". More abstractly, we might say that modality affects the circumstances in which we take an assertion to be satisfied.
Informal reasoning
Main article:
Informal logic
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.^{[23]}
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.
^{[24]} Many other ancient and medieval philosophers applied mathematical ideas and methods to their philosophical claims.
^{[25]}
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.
^{[26]} Despite the negative nature of the incompleteness theorems,
Gödel's completeness theorem, a result in
model theory and another application of mathematics to logic, can be understood as showing how close logicism came to being true: every rigorously defined mathematical theory can be exactly captured by a firstorder logical theory; Frege's
proof calculus is enough to
describe the whole of mathematics, though not
equivalent to it. Thus we see how complementary the two areas of mathematical logic have been.
^{[citation needed]}
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 nonstandard logics (e.g.,
free logics,
tense logics) as well as various extensions of
classical logic (e.g.,
modal logics), and nonstandard semantics for such logics (e.g.,
Kripke's technique of supervaluations in the semantics of logic).
Logic and the philosophy of language are closely related. Philosophy of language has to do with the study of how our language engages and interacts with our thinking. Logic has an immediate impact on other areas of study. Studying logic and the relationship between logic and ordinary speech can help a person better structure their own arguments and critique the arguments of others. Many popular arguments are filled with errors because so many people are untrained in logic and unaware of how to correctly formulate an argument.
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.
Furthermore, computers can be used as tools for logicians. For example, in symbolic logic and mathematical logic, proofs by humans can be computerassisted. Using
automated theorem proving the machines can find and check proofs, as well as work with proofs too lengthy to be written out by hand.
Controversies in logic
Just as we have seen there is disagreement over what logic is about, so there is disagreement about what logical truths there are.
Bivalence and the law of the excluded middle
The logics discussed above are all "
bivalent" or "twovalued"; that is, they are most naturally understood as dividing propositions into true and false propositions.
Nonclassical 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".
^{[29]}
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.
^{[citation needed]} In the early 20th century
Jan Łukasiewicz investigated the extension of the traditional true/false values to include a third value, "possible", so inventing
ternary logic, the first
multivalued logic.
^{[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.
^{[30]}
Modal logic is not truth conditional, and so it has often been proposed as a nonclassical logic. However, modal logic is normally formalised with the principle of the excluded middle, and its
relational semantics is bivalent, so this inclusion is disputable.
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?"
^{[31]} 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.
^{[32]}
Another paper by the same name by
Sir Michael Dummett argues that Putnam's desire for realism mandates the law of distributivity.
^{[33]} Distributivity of logic is essential for the realist's understanding of how propositions are true of the world in just the same way as he has argued the principle of bivalence is. In this way, the question, "Is logic empirical?" can be seen to lead naturally into the fundamental controversy in
metaphysics on
realism versus antirealism.
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 NonContradiction. It was based on
Leibniz's idea that this law of logic also requires a sufficient ground in order 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 noncontradiction, of identity, itself relies upon difference and so is not independently assertable.
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^{[35]}. 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^{[36]}. 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^{[37]}.
See also
Notes
 ^ "possessed of reason, intellectual, dialectical, argumentative", also related to λόγος (logos), "word, thought, idea, argument, account, reason, or principle" (Liddell & Scott 1999; Online Etymology Dictionary 2001).
 ^ Welton, James (1896). A manual of logic. University tutorial series. 1 (2nd ed.). W.B. Clive. http://books.google.com/books?id=KaAZAAAAMAAJ&pg=PA12&lpg=PA12&dq=%22art+and+science+of+reasoning%22.
 ^ ^{a} ^{b} Hofweber, T. (2004). "Logic and Ontology". in Zalta, Edward N. Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/logicontology.
 ^ Averroes, In Arist. Physicam I, textus 35, ed. Juntina, IV, fol. 11vb.
 ^ Cox, J. Robert; Willard, Charles Arthur, eds (1983). Advances in Argumentation Theory and Research. Southern Illinois University Press. ISBN 9780809310500.
 ^ Plato (1976). Buchanan, Scott. ed. The Portable Plato. Penguin. ISBN 0140150404.
 ^ Aristotle (2001). "Posterior Analytics". in Mckeon, Richard. The Basic Works. Modern Library. ISBN 0375757996.
 ^ ^{a} ^{b} ^{c} Whitehead, Alfred North; Russell, Bertrand (1967). Principia Mathematica to *56. Cambridge University Press. ISBN 0521626064.
 ^ ^{a} ^{b} For a more modern treatment, see Hamilton, A. G. (1980). Logic for Mathematicians. Cambridge University Press. ISBN 0521292913.
 ^ Łukasiewicz, Jan (1957). Aristotle's syllogistic from the standpoint of modern formal logic (2nd ed.). Oxford University Press. p. 7. ISBN 9780198241447.
 ^ ^{a} ^{b} ^{c} Bergmann, Merrie, James Moor, and Jack Nelson. The Logic Book fifth edition. New York, NY: McGrawHill, 2009.
 ^ Mendelson, Elliott (1964). "Quantification Theory: Completeness Theorems". Introduction to Mathematical Logic. Van Nostrand. ISBN 0412808307.
 ^ E.g., Kline (1972, p.53) wrote "A major achievement of Aristotle was the founding of the science of logic".
 ^ Kak, S. (2004). The Architecture of Knowledge. Delhi: CSC.
 ^ The four Catuṣkoṭi logical divisions are formally very close to the four opposed propositions of the Greek tetralemma, which in turn are analogous to the four truth values of modern relevance logic Cf. Belnap (1977); Jayatilleke, K. N., (1967, The logic of four alternatives, in Philosophy East and West, University of Hawaii Press).
 ^ "School of Names". Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/schoolnames/. Retrieved 5 September 2008.
 ^ Goodman, Lenn Evan (1992). Avicenna. Routledge. p. 184. ISBN 9780415019293.
 ^ Goodman, Lenn Evan (2003). Islamic Humanism. Oxford University Press. p. 155. ISBN 0195135806.
 ^ "History of logic: Arabic logic". Encyclopædia Britannica. http://www.britannica.com/EBchecked/topic/346217/historyoflogic/65928/Arabiclogic.
 ^ Nabavi, Lotfollah. "Sohrevardi's Theory of Decisive Necessity and kripke's QSS System". Journal of Faculty of Literature and Human Sciences. Archived from the original on 26 January 2008. http://web.archive.org/web/20080126100838/http://public.ut.ac.ir/html/fac/lit/articles.html.
 ^ "Science and Muslim Scientists". Islam Herald. Archived from the original on 17 December 2007. http://web.archive.org/web/20071217150016/http://www.islamherald.com/asp/explore/science/science_muslim_scientists.asp.
 ^ Hallaq, Wael B. (1993). Ibn Taymiyya Against the Greek Logicians. Oxford University Press. p. 48. ISBN 0198240430.
 ^ Stolyar, Abram A. (1983). Introduction to Elementary Mathematical Logic. Dover Publications. p. 3. ISBN 0486645614.
 ^ Barnes, Jonathan (1995). The Cambridge Companion to Aristotle. Cambridge University Press. p. 27. ISBN 0521422949.
 ^ Aristotle (1989). Prior Analytics. Hackett Publishing Co.. p. 115. ISBN 9780872200647.
 ^ Mendelson, Elliott (1964). "Formal Number Theory: Gödel's Incompleteness Theorem". Introduction to Mathematical Logic. Monterey, Calif.: Wadsworth & Brooks/Cole Advanced Books & Software. OCLC 13580200.
 ^ Brookshear, J. Glenn (1989). "Computability: Foundations of Recursive Function Theory". Theory of computation: formal languages, automata, and complexity. Redwood City, Calif.: Benjamin/Cummings Pub. Co.. ISBN 0805301437.
 ^ Brookshear, J. Glenn (1989). "Complexity". Theory of computation: formal languages, automata, and complexity. Redwood City, Calif.: Benjamin/Cummings Pub. Co.. ISBN 0805301437.
 ^ Hegel, G. W. F (1971) [1817]. Philosophy of Mind. Encyclopedia of the Philosophical Sciences. trans. William Wallace. Oxford: Clarendon Press. p. 174. ISBN 0198750145.
 ^ Hájek, Petr (2006). "Fuzzy Logic". in Zalta, Edward N.. Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/logicfuzzy/.
 ^ Putnam, H. (1969). "Is Logic Empirical?". Boston Studies in the Philosophy of Science 5.
 ^ Birkhoff, G.; von Neumann, J. (1936). "The Logic of Quantum Mechanics". Annals of Mathematics 37: 823–843.
 ^ Dummett, M. (1978). "Is Logic Empirical?". Truth and Other Enigmas. ISBN 0674910761.
 ^ Priest, Graham (2008). "Dialetheism". in Zalta, Edward N.. Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/dialetheism.
 ^ Nietzsche, 1873, On Truth and Lies in a Nonmoral Sense.
 ^ Nietzsche, 1882, The Gay Science.
 ^ Nietzsche, 1878, Human, All Too Human
References
 Nuel Belnap, (1977). A useful fourvalued logic. In Dunn & Eppstein, Modern uses of multiplevalued logic. Reidel: Boston.
 Brookshear, J. Glenn (1989). Theory of computation: formal languages, automata, and complexity. Redwood City, Calif.: Benjamin/Cummings Pub. Co.. ISBN 0805301437.
 Cohen, R.S, and Wartofsky, M.W. (1974). Logical and Epistemological Studies in Contemporary Physics. Boston Studies in the Philosophy of Science. D. Reidel Publishing Company: Dordrecht, Netherlands. ISBN 9027703779.
 Finkelstein, D. (1969). "Matter, Space, and Logic". in R.S. Cohen and M.W. Wartofsky (eds. 1974).
 Gabbay, D.M., and Guenthner, F. (eds., 2001–2005). Handbook of Philosophical Logic. 13 vols., 2nd edition. Kluwer Publishers: Dordrecht.
 Hilbert, D., and Ackermann, W, (1928). Grundzüge der theoretischen Logik (Principles of Mathematical Logic). SpringerVerlag. OCLC 2085765
 Susan Haack, (1996). Deviant Logic, Fuzzy Logic: Beyond the Formalism, University of Chicago Press.
 Hodges, W., (2001). Logic. An introduction to Elementary Logic, Penguin Books.
 Hofweber, T., (2004), Logic and Ontology. Stanford Encyclopedia of Philosophy. Edward N. Zalta (ed.).
 Hughes, R.I.G., (1993, ed.). A Philosophical Companion to FirstOrder Logic. Hackett Publishing.
 Kline, Morris (1972). Mathematical Thought From Ancient to Modern Times. Oxford University Press. ISBN 0195061357.
 Kneale, William, and Kneale, Martha, (1962). The Development of Logic. Oxford University Press, London, UK.
 Liddell, Henry George; Scott, Robert. "Logikos". A GreekEnglish Lexicon. Perseus Project. http://www.perseus.tufts.edu/cgibin/ptext?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3D%2363716. Retrieved 8 May 2009.
 Mendelson, Elliott, (1964). Introduction to Mathematical Logic. Wadsworth & Brooks/Cole Advanced Books & Software: Monterey, Calif. OCLC 13580200
 Harper, Robert (2001). "Logic". Online Etymology Dictionary. http://www.etymonline.com/index.php?term=logic. Retrieved 8 May 2009.
 Smith, B., (1989). "Logic and the Sachverhalt". The Monist 72(1):52–69.
 Whitehead, Alfred North and Bertrand Russell, (1910). Principia Mathematica. Cambridge University Press: Cambridge, England. OCLC 1041146
External links and further reading
Logic 

Related articles 

Academic areas 


Foundational concepts 
















Portal · Category · Outline · WikiProject · Talk · changes 
