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Logic (from the Ancient Greek: λογική, translit. logikḗ[1]), originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason", is a subject concerned with the most general laws of truth,[2] and is now generally held to consist of the systematic study of the form of valid inference. A valid inference is one where there is a specific relation of logical support between the assumptions of the inference and its conclusion. (In ordinary discourse, inferences may be signified by words like therefore, hence, ergo, and so on.)

There is no universal agreement as to the exact scope and subject matter of logic (see § Rival conceptions, below), but it has traditionally included the classification of arguments, the systematic exposition of the 'logical form' common to all valid arguments, the study of inference, including fallacies, and the study of semantics, including paradoxes. Historically, logic has been studied in philosophy (since ancient times) and mathematics (since the mid-19th century), and recently logic has been studied in computer science, linguistics, psychology, and other fields.


The concept of logical form is central to logic. 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 logic.

  • Informal logic is the study of natural language arguments. The study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, informal logic is not logic at all. See 'Rival conceptions', below.
  • Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference.[4][5] Symbolic logic is often divided into two main branches: propositional logic and predicate logic.

However, agreement on what logic is has remained elusive, and although the field of universal logic has studied the common structure of logics, in 2007 Mossakowski et al. commented that "it is embarrassing that there is no widely acceptable formal definition of 'a logic'".[6]

Logical form[edit]

Logic is generally considered formal when it analyzes and represents the 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. Simply put, to formalize simply means to translate English sentences into the language of logic.


Second, certain parts of the sentence must be replaced with schematic letters. Thus, for example, the expression "all Ps are Qs" shows the logical form common to the sentences "all men are mortals", "all cats are carnivores", "all Greeks are philosophers", and so on. The schema can further be condensed into the formula A(P,Q), where the letter A indicates the judgement 'all - are -'.

The importance of form was recognised from 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".[7] According to the followers of Aristotle (such as Ammonius), only the logical principles stated in schematic terms belong to logic, not those given in concrete terms. The concrete terms "man", "mortal", etc., are analogous to the substitution values of the schematic placeholders P, Q, R, which were called the "matter" (Greek hyle) of the inference.

There is a big difference between the kinds of formulas seen in traditional term logic and the predicate calculus that is the fundamental advance of modern logic. The formula A(P,Q) (all Ps are Qs) of traditional logic corresponds to the more complex formula in predicate logic, involving the logical connectives for universal quantification and implication rather than just the predicate letter A and using variable arguments where traditional logic uses just the term letter P. With the complexity comes power, and the advent of the predicate calculus inaugurated revolutionary growth of the subject.


The validity of an argument depends upon the meaning or semantics of the sentences that make it up.

Aristotle's Organon, especially On Interpretation, gives a cursory outline of semantics which the scholastic logicians, particularly in the thirteenth and fourteenth century, developed into a complex and sophisticated theory, called Supposition Theory. This showed how the truth of simple sentences, expressed schematically, depend on how the terms 'supposit' or stand for certain extra-linguistic items. For example, in part II of his Summa Logicae, William of Ockham presents a comprehensive account of the necessary and sufficient conditions for the truth of simple sentences, in order to show which arguments are valid and which are not. Thus "every A is B' is true if and only if there is something for which 'A' stands, and there is nothing for which 'A' stands, for which 'B' does not also stand." [8]

Early modern logic defined semantics purely as a relation between ideas. Antoine Arnauld in the Port Royal Logic, says that 'after conceiving things by our ideas, we compare these ideas, and, finding that some belong together and some do not, we unite or separate them. This is called affirming or denying, and in general judging.[9] Thus truth and falsity are no more than the agreement or disagreement of ideas. This suggests obvious difficulties, leading Locke to distinguish between 'real' truth, when our ideas have 'real existence' and 'imaginary' or 'verbal' truth, where ideas like harpies or centaurs exist only in the mind.[10] This view (psychologism) was taken to the extreme in the nineteenth century, and is generally held by modern logicians to signify a low point in the decline of logic before the twentieth century.

Modern semantics is in some ways closer to the medieval view, in rejecting such psychological truth-conditions. However, the introduction of quantification, needed to solve the problem of multiple generality, rendered impossible the kind of subject-predicate analysis that underlies medieval semantics. The main modern approach is model-theoretic semantics, based on Alfred Tarski's semantic theory of truth. The approach assumes that the meaning of the various parts of the propositions are given by the possible ways we can give a recursively specified group of interpretation functions from them to some predefined domain of discourse: an interpretation of first-order predicate logic is given by a mapping from terms to a universe of individuals, and a mapping from propositions to the truth values "true" and "false". Model-theoretic semantics is one of the fundamental concepts of model theory. Modern semantics also admits rival approaches, such as the proof-theoretic semantics that associates the meaning of propositions with the roles that they can play in inferences, an approach that ultimately derives from the work of Gerhard Gentzen on structural proof theory and is heavily influenced by Ludwig Wittgenstein's later philosophy, especially his aphorism "meaning is use".


Inference is not to be confused with implication. An implication is a sentence of the form 'If p then q', and can be true or false. The Stoic logician Philo of Megara was the first to define the truth conditions of such an implication: false only when the antecedent p is true and the consequent q is false, in all other cases true. An inference, on the other hand, consists of two separately asserted propositions of the form 'p therefore q'. An inference is not true or false, but valid or invalid. However, there is a connection between implication and inference, as follows: if the implication 'if p then q' is true, the inference 'p therefore q' is valid. This was given an apparently paradoxical formulation by Philo, who said that the implication 'if it is day, it is night' is true only at night, so the inference 'it is day, therefore it is night' is valid in the night, but not in the day.

The theory of inference (or 'consequences') was systematically developed in medieval times by logicians such as William of Ockham and Walter Burley. It is uniquely medieval, though it has its origins in Aristotle's Topics and Boethius' De Syllogismis hypotheticis. This is why many terms in logic are Latin. For example, the rule that licenses the move from the implication 'if p then q' plus the assertion of its antecedent p, to the assertion of the consequent q is known as modus ponens (or 'mode of positing'). Its Latin formulation is 'Posito antecedente ponitur consequens'. The Latin formulations of many other rules such as 'ex falso quodlibet' (anything follows from a falsehood), 'reductio ad absurdum' (disproof by showing the consequence is absurd) also date from this period.

However, the theory of consequences, or of the so-called 'hypothetical syllogism' was never fully integrated into the theory of the 'categorical syllogism'. This was partly because of the resistance to reducing the categorical judgment 'Every S is P' to the so-called hypothetical judgment 'if anything is S, it is P'. The first was thought to imply 'some S is P', the second was not, and as late as 1911 in the Encyclopædia Britannica article on Logic, we find the Oxford logician T.H. Case arguing against Sigwart's and Brentano's modern analysis of the universal proposition.

Logical systems[edit]

A formal system is an organization of terms used for the analysis of deduction. It consists of an alphabet, a language over the alphabet to construct sentences, and a rule for deriving sentences. Among the important properties that logical systems can have are:

  • Completeness, which means that if a formula is true, it can be proven, i.e. is a theorem of the system.
  • Soundness, meaning that if any formula is a theorem of the system, it is true. This is the converse of completeness. (Note that in a distinct philosophical use of the term, an argument is sound when it is both valid and its premises are true).[12]

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;[5] however, first-order predicate logics not extended by specific axioms to be arithmetic formal systems with equality can be complete and consistent.[13]

Logic and rationality[edit]

As the study of argument is of clear importance to the reasons that we hold things to be true, logic is of essential importance to rationality. Here we have defined logic to be "the systematic study of the form of arguments"; the reasoning behind argument is of several sorts, but only some of these arguments fall under the aegis of logic proper.

Deductive reasoning concerns the logical consequence of given premises and is the form of reasoning most closely connected to logic. On a narrow conception of logic (see below) logic concerns just deductive reasoning, although such a narrow conception controversially excludes most of what is called informal logic from the discipline.

There are other forms of reasoning that are rational but that are generally not taken to be part of logic. These include inductive reasoning, which covers forms of inference that move from collections of particular judgements to universal judgements, and abductive reasoning,[14] which is a form of inference that goes from observation to a hypothesis that accounts for the reliable data (observation) and seeks to explain relevant evidence. The American philosopher Charles Sanders Peirce (1839–1914) first introduced the term as "guessing".[15] Peirce said that to abduce a hypothetical explanation from an observed surprising circumstance is to surmise that may be true because then would be a matter of course.[16] Thus, to abduce from involves determining that is sufficient (or nearly sufficient), but not necessary, for .

While inductive and abductive inference are not part of logic proper, the methodology of logic has been applied to them with some degree of success. For example, the notion of deductive validity (where 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) exists in an analogy to the notion of inductive validity, or "strength", where an inference is inductively strong if and only if its premises give some degree of probability to its conclusion. Whereas 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 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 logical association rule induction, while others may use mathematical models of probability such as decision trees.

Rival conceptions[edit]


Logic has been defined[by whom?] as "the study of arguments correct in virtue of their form". This has not been the definition taken in this article, but the idea that logic treats special forms of argument, deductive argument, rather than argument in general, has a history in logic that dates back at least to logicism in mathematics (19th and 20th centuries) and the advent of the influence of mathematical logic on philosophy. A consequence of taking logic to treat special kinds of argument is that it leads to identification of special kinds of truth, the logical truths (with logic equivalently being the study of logical truth), and excludes many of the original objects of study of logic that are treated as informal logic. Robert Brandom has argued against the idea that logic is the study of a special kind of logical truth, arguing that instead one can talk of the logic of material inference (in the terminology of Wilfred Sellars), with logic making explicit the commitments that were originally implicit in informal inference.[18][page needed]


Aristotle, 384–322 BCE.

In Europe, logic was first developed by Aristotle.[19] Aristotelian logic became widely accepted in science and mathematics and remained in wide use in the West until the early 19th century.[20] Aristotle's system of logic was responsible for the introduction of hypothetical syllogism,[21] temporal modal logic,[22][23] and inductive logic,[24] as well as influential terms such as terms, predicables, syllogisms and propositions. 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, often using variations of the methodology of scholasticism. In 1323, William of Ockham's influential Summa Logicae was released. By the 18th century, the structured approach to arguments had degenerated and fallen out of favour, as depicted in Holberg's satirical play Erasmus Montanus.

The Chinese logical philosopher Gongsun Long (c. 325–250 BCE) proposed the paradox "One and one cannot become two, since neither becomes two."[25] 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, the Anviksiki school of logic was founded by Medhatithi Gautama (c. 6th century BCE).[26] 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.[27] 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.[28] In the 20th century, Western philosophers like Stanislaw Schayer and Klaus Glashoff have explored Indian logic more extensively.


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.


Syllogistic logic[edit]

A depiction from the 15th century of the square of opposition, which expresses the fundamental dualities of syllogistic.




I was upset. I had always believed logic was a universal weapon, and now I realized how its validity depended on the way it was employed.[30]

Propositional logic[edit]

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 establishes certain formulae as "theorems". An example of a theorem of propositional logic is , which says that if A holds, then B implies A.

Predicate logic[edit]

Gottlob Frege's Begriffschrift introduced the notion of quantifier in a graphical notation, which here represents the judgement that is true.

Predicate logic is the generic term for symbolic formal systems such as first-order logic, second-order logic, many-sorted logic, and infinitary logic. It provides an account of quantifiers general enough to express a wide set of arguments occurring in natural language. For example, Bertrand Russell's famous barber paradox, "there is a man who shaves all and only men who do not shave themselves" can be formalised by the sentence , using the non-logical predicate to indicate that x is a man, and the non-logical relation to indicate that x shaves y; all other symbols of the formulae are logical, expressing the universal and existential quantifiers, conjunction, implication, negation and biconditional.

Whilst 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—allowing predicate logic to solve the problem of multiple generality that had perplexed medieval logicians.



Modal logic[edit]


Aristotle's logic is in large parts concerned with the theory of non-modalized logic. Although, there are passages in his work, such as the famous sea-battle argument in De Interpretatione § 9, that are now seen as anticipations of modal logic and its connection with potentiality and time, the earliest formal system of modal logic was developed by Avicenna, who ultimately developed a theory of "temporally modalized" syllogistic.[31]


Informal reasoning and dialectic[edit]


This ancient motivation is still alive, although it no longer takes centre stage in the picture of logic; typically dialectical logic forms the heart of a course in critical thinking, a compulsory course at many universities. Dialectic has been linked to logic since ancient times, but it has not been until recent decades that European and American logicians have attempted to provide mathematical foundations for logic and dialectic by formalising dialectical logic. Dialectical logic is also the name given to the special treatment of dialectic in Hegelian and Marxist thought. There have been pre-formal treatises on argument and dialectic, from authors such as Stephen Toulmin (The Uses of Argument), Nicholas Rescher (Dialectics),[32][33][34] and van Eemeren and Grootendorst (Pragma-dialectics). Theories of defeasible reasoning can provide a foundation for the formalisation of dialectical logic and dialectic itself can be formalised as moves in a game, where an advocate for the truth of a proposition and an opponent argue. Such games can provide a formal game semantics for many logics.


Mathematical logic[edit]


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.[36] Many other ancient and medieval philosophers applied mathematical ideas and methods to their philosophical claims.[37]

One of the boldest attempts to apply logic to mathematics was the logicism pioneered by philosopher-logicians such as Gottlob Frege and Bertrand Russell. Mathematical theories were supposed to be logical tautologies, and the programme was to show this by means of a reduction of mathematics to logic.[4] The various attempts to carry this out met with failure, 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.



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.[40] 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.[41]

Philosophical logic[edit]



Computational logic[edit]

A simple toggling circuit is expressed using a logic gate and a synchronous register.



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:

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 write out by hand.

Non-classical logic[edit]

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 that reject various rules of Classical logic.


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.[45] 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 in the Western tradition.[46]


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 formalization in mathematics, but his student Arend Heyting studied intuitionistic logic formally, as did Gerhard Gentzen. Intuitionistic logic is of great interest to computer scientists, as it is a constructive logic and sees many applications, such as extracting verified programs from proofs and influencing the design of programming languages through the formulae-as-types correspondence.



"Is Logic Empirical?"[edit]

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?"[48] 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.[49]

Another paper of the same name by Michael Dummett argues that Putnam's desire for realism mandates the law of distributivity.[50] 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[edit]




Tolerating the impossible[edit]



Rejection of logical truth[edit]

The philosophical vein of various kinds of skepticism contains many kinds of doubt and rejection of the various bases on 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. This is in contrast with the usual views in philosophical skepticism, where logic directs skeptical enquiry to doubt received wisdoms, as in the work of Sextus Empiricus.


This position held by Nietzsche however, has come under extreme scrutiny for several reasons. Some philosophers, such as Jürgen Habermas, claim his position is self-refuting—and accuse Nietzsche of not even having a coherent perspective, let alone a theory of knowledge.[55] Georg Lukács, in his book The Destruction of Reason, asserts 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."[56] Bertrand Russell described Nietzsche's irrational claims with "He is fond of expressing himself paradoxically and with a view to shocking conventional readers" in his book A History of Western Philosophy.[57]

See also[edit]

Notes and references[edit]

  2. ^ Due to Frege, see the Definitions of logic article.
  4. ^ a b c Whitehead, Alfred North; Russell, Bertrand (1967). Principia Mathematica to *56. Cambridge University Press. ISBN 0-521-62606-4. 
  7. ^ Łukasiewicz, Jan (1957). Aristotle's syllogistic from the standpoint of modern formal logic (2nd ed.). Oxford University Press. p. 7. ISBN 978-0-19-824144-7. 
  8. ^ Summa Logicae Part II c.4 transl. as Ockam's Theory of Propositions, A. Freddoso and H. Schuurman, St Augustine's Press 1998, p. 96
  9. ^ Arnauld, Logic or the Art of Thinking Part 2 Chapter 3.
  10. ^ Locke, 1690. An Essay Concerning Human Understanding, IV. v. 1-8)
  11. ^ Bergmann, Merrie; Moor, James; Nelson, Jack (2009). The Logic Book (Fifth ed.). New York, NY: McGraw-Hill. ISBN 978-0-07-353563-0. 
  12. ^ Internet Encyclopedia of Philosophy, Validity and Soundness
  13. ^ Mendelson, Elliott (1964). "Quantification Theory: Completeness Theorems". Introduction to Mathematical Logic. Van Nostrand. ISBN 0-412-80830-7. 
  14. ^ On abductive reasoning, see:
    ISBN 1-58619-794-2 Parameter error in {{isbn}}: Invalid ISBN.
  15. ^ See Abduction and Retroduction at Commens Dictionary of Peirce's Terms, and see Peirce's papers:
  18. ^ Brandom, Robert (2000). Articulating Reasons. Cambridge, MA: Harvard University Press. ISBN 0-674-00158-3. 
  20. ^ "Aristotle", MTU Department of Chemistry.
  27. ^ Kisor Kumar Chakrabarti (June 1976). "Some Comparisons Between Frege's Logic and Navya-Nyaya Logic". Philosophy and Phenomenological Research. International Phenomenological Society. 36 (4): 554–563. doi:10.2307/2106873. JSTOR 2106873. This paper consists of three parts. The first part deals with Frege's distinction between sense and reference of proper names and a similar distinction in Navya-Nyaya logic. In the second part we have compared Frege's definition of number to the Navya-Nyaya definition of number. In the third part we have shown how the study of the so-called 'restrictive conditions for universals' in Navya-Nyaya logic anticipated some of the developments of modern set theory. 
  28. ^ Jonardon Ganeri (2001). Indian logic: a reader. Routledge. pp. vii, 5, 7. ISBN 0-7007-1306-9. 
  29. ^ "Aristotle". Encyclopædia Britannica. 
  30. ^ Eco, Umberto (1980). The Name of the Rose. London: Vintage. p. 253. ISBN 9780099466031. 
  31. ^ "History of logic: Arabic logic". Encyclopædia Britannica. 
  32. ^ Rescher, Nicholas (1978). "Dialectics: A Controversy-Oriented Approach to the Theory of Knowledge". Informal Logic. 1 (#3). 
  33. ^ Hetherington, Stephen (2006). "Nicholas Rescher: Philosophical Dialectics". Notre Dame Philosophical Reviews (2006.07.16). 
  34. ^ Rescher, Nicholas (2009). Jacquette,Dale, ed. Reason, Method, and Value: A Reader on the Philosophy of Nicholas Rescher. Ontos Verlag. ISBN 3110329050. 
  37. ^ Aristotle (1989). Prior Analytics. Hackett Publishing Co. p. 115. ISBN 978-0-87220-064-7. 
  39. ^ Barwise (1982) divides the subject of mathematical logic into model theory, proof theory, set theory and recursion theory.
  40. ^ 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 0-8053-0143-7. 
  42. ^ Goldman, Alvin I. (1986), Epistemology and Cognition, Harvard University Press, p. 293, ISBN 9780674258969, untrained subjects are prone to commit various sorts of fallacies and mistakes .
  43. ^ Demetriou, A.; Efklides, A., eds. (1994), Intelligence, Mind, and Reasoning: Structure and Development, Advances in Psychology, 106, Elsevier, p. 194, ISBN 9780080867601 .
  44. ^ Hegel, G. W. F (1971) [1817]. Philosophy of Mind. Encyclopedia of the Philosophical Sciences. trans. William Wallace. Oxford: Clarendon Press. p. 174. ISBN 0-19-875014-5. 
  45. ^ Joseph E. Brenner (3 August 2008). Logic in Reality. Springer. pp. 28–30. ISBN 978-1-4020-8374-7. Retrieved 9 April 2012. 
  46. ^ Zegarelli, Mark (2010), Logic For Dummies, John Wiley & Sons, p. 30, ISBN 9781118053072 .
  47. ^ Hájek, Petr (2006). "Fuzzy Logic". In Zalta, Edward N. Stanford Encyclopedia of Philosophy. 
  48. ^ Putnam, H. (1969). "Is Logic Empirical?". Boston Studies in the Philosophy of Science. 5. 
  49. ^ Birkhoff, G.; von Neumann, J. (1936). "The Logic of Quantum Mechanics". Annals of Mathematics. Annals of Mathematics. 37 (4): 823–843. doi:10.2307/1968621. JSTOR 1968621. 
  51. ^ Priest, Graham (2008). "Dialetheism". In Zalta, Edward N. Stanford Encyclopedia of Philosophy. 
  52. ^ Nietzsche, 1873, On Truth and Lies in a Nonmoral Sense.
  53. ^ Nietzsche, 1882, The Gay Science.
  54. ^ Nietzsche, 1878, Human, All Too Human
  55. ^ Babette Babich, Habermas, Nietzsche, and Critical Theory
  56. ^ Georg Lukács. "The Destruction of Reason by Georg Lukács 1952". Marxists.org. Retrieved 2013-06-16. 
  57. ^ Russell, Bertrand (1945), A History of Western Philosophy And Its Connection with Political and Social Circumstances from the Earliest Times to the Present Day (PDF), Simon and Schuster, p. 762, archived from the original on 28 May 2014 


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