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Draft:Original research/Logic

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In 1869, as women were not granted much access to laboratories and observatories, Christine Ladd-Franklin turned to mathematics and logic, which did not require any apparatus. Credit: Smithsonian Institution.

Logic is more than reasoning. Usually it is reasoning conducted or assessed according to strict principles of validity. Aristotelian logic is a particular system or codification of the principles of proof and inference.

At a secondary level an introduction to logic may be helpful, where some of the more common operators are described. This introduction is a part of elementary logic at the undergraduate level. Here, there is at least one lesson available.

This learning resource is partly an article, in some subareas an essay, and mostly a lecture.

Logic is often considered a part of philosophy. And, most often is used in science to help create knowledge consisting of facts and truths. But, it finds needed applicability in law and the practice of law. A third popular field that confers a rigid structure on logic is mathematics.

Nearly all efforts, intellectual or otherwise, can be approached and have some understanding produced through the application of logic. This includes volition (e.g., emotion), affections, morality, and religion.

Reasoning

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Main resource: Reasoning

Logic can also mean the quality of being justifiable by reason.

Reason is the capacity of consciously making sense of things, establishing and verifying facts, applying logic, and changing or justifying practices, institutions, and beliefs based on new or existing information.[1] It is closely associated with such characteristically human activities as philosophy, science, language, mathematics and art, and is normally considered to be a distinguishing ability possessed by humans.[2]

"[...] the exercise of independent practical reasoning is one essential constituent to full human flourishing."[2]

Def. the "deduction of inferences or interpretations from premises"[3] or "the drawing of inferences or conclusions through the use of [...] statement[s] offered in explanation or justification"[4] is called reasoning.

The philosophical field of logic studies ways in which humans reason formally through argument.[5]

A distinction is often drawn between logical, discursive reasoning (reason proper), and intuitive reasoning.[6]

Reasoning, as a part of executive decision making, is also closely identified with the ability to self-consciously change, in terms of goals, beliefs, attitudes, traditions, and institutions, and therefore with the capacity for freedom and self-determination.[7]

In contrast to the use of "reason" as an abstract noun, a reason is a consideration given which either explains or justifies events, phenomena, or behavior.[8]

When evaluating a moral decision, "morality is, at the very least, the effort to guide one's conduct by reason—that is, doing what there are the best reasons for doing—while giving equal [and impartial] weight to the interests of all those affected by what one does."[9]

Reason is often said to be reflexive, or "self-correcting", and the critique of reason has been a persistent theme in philosophy.[10]

Practical reasoning is the self-legislating or self-governing formulation of universal norms, and theoretical reasoning the way humans posit universal laws of nature.[11]

Under practical reason, the moral autonomy or freedom of human beings depends on their ability to behave according to laws that are given to them by the proper exercise of that reason, contrasted with earlier forms of morality, which depended on religious understanding and interpretation, or nature for their substance.[12]

In a free society each individual must be able to pursue their goals however they see fit, so long as their actions conform to principles given by reason, called the "categorical imperative", which would justify an action only if it could be universalized:

"Act only according to that maxim whereby you can, at the same time, will that it should become a universal law.[13]

Philosophy

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Main resource: Philosophy
 
This is a photo of the French philosopher Catherine Perret. Credit: Valueyou.

Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language.[14][15] Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational argument.[16]

"Philosophy is a study of problems which are ultimate, abstract and general. These problems are concerned with the nature of existence, knowledge, morality, reason and human purpose."[14]

"The aim of philosophical inquiry is to gain insight into questions about knowledge, truth, reason, reality, meaning, mind, and value."[15]

Theoretical logic

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“[D]efinitions are always of symbols, for only symbols have meanings for definitions to explain.”[17] A term can be one or more of a set of symbols such as words, phrases, letter designations, or any already used symbol or new symbol.

In the theory of definition, “the symbol being defined is called the definiendum, and the symbol or set of symbols used to explain the meaning of the definiendum is called the definiens.”[17] “The definiens is not the meaning of the definiendum, but another symbol or group of symbols which, according to the definition, has the same meaning as the definiendum.”[17]

Def.

a(1): "a science that deals with the canons and criteria of validity of inference and demonstration : the science of the normative formal principles of reasoning",
(2): "a branch of semiotic; [especially: syntactics]",
(3): "the formal principles of a branch of knowledge",
b: "a particular mode of reasoning", and
c: "interrelation or sequence of facts or events when seen as inevitable or predictable"

is called logic.[4]

Similar to the above dictionary, or lexical, definition is

Def. "[l]ogic is the study of correct argumentation."[18]

Def. a "method of human thought that involves thinking in a linear, step-by-step manner about how a problem can be solved"[19] is called logic.

Logicism

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"The logistic thesis is that mathematics is a branch of logic. The mathematical notions are to be defined in terms of the logical notions. The theorems of mathematics are to be proved as theorems of logic."[20]

Def. a "generalized kind of number used to denote the size of a set, including infinite sets"[21] is called a cardinal number.

Def. "a cardinal number which possesses every property P such that

  1. 0 has the property P and
  2. n + 1 has the property P whenever n has the property P"[20] is called a finite cardinal (or natural number).

Def. "a non-negative integer [{0, 1, 2, ...}][22]"[23] is called a natural number.

Def. "a cardinal number for which mathematical induction holds"[20] is called a natural number.

One way to "adapt the logicistic construction of mathematics to the situation arising from the discovery of the paradoxes"[20] is to exclude "impredicative definitions".[20]

Ramified theory of types

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"The primary objects or individuals (i.e. the given things not being subjected to logical analysis) are assigned to one type (say type 0), properties of individuals to type 1, properties of properties of individuals to type 2, etc.; and no properties are admitted which do not fall into one of these logical types (e.g. this puts the properties 'predicable' and 'impredicable' [...] outside the pale of logic)."[20]

Relations "and classes [would be some] admitted types for other objects".[20] To "exclude impredicative definition within a type, the types above type 0 are further separated into orders. Thus for type 1, properties defined without mentioning any totality belong to order 0, and properties defined using the totality of properties of a given order belong to the next higher order. (The logicistic definition of natural number now becomes predicative, when the P in it is specified to range only over properties of a given order, in which case the property of being a natural number is of the next higher order.)"[20]

Axiom of reducibility

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To escape that the above "separation into orders makes it impossible to construct the familiar analysis [may require an] axiom of reducibility, which asserts that to any property belonging to an order above the lowest, there is a coextensive property (i.e. one possessed by exactly the same objects) of order 0. If only definable properties are considered to exist, then the axiom means that to every impredicative definition within a given type there is an equivalent predictive one."[20]

The "difficulty is [...]: on what grounds shall we believe in the axiom of reducibility? If properties are to be constructed, the matter should be settled on the basis of constructions, not by an axiom."[20]

"This axiom has a purely pragmatic justification: it leads to the desired results, and to no others [so far as is known]. But clearly it is not the sort of axiom with which we can rest content."[24]

The "desired result and no others can apparently be obtained without the hierarchy of orders (i.e. with a simple theory of types)."[20]

The paradoxes can be classified "into two sorts,

  1. 'logical' (e.g. Burali-Forti, Cantor and Russell) and
  2. 'epistemological' or 'semantical' (e.g. the Richard and Epimenides); and [...] the logical antinomies are (apparently) stopped by the simple hierarchy of types, and the semantical ones are (apparently) prevented from arising within the symbolic language by the absence therein of the requisite means for referring to expressions of the same language."[20]

Arguments "to justify impredicative definitions within a type entail a conception of the totality of predicates of the type as existing independently of their constructibility or definability."[20]

"Logicism treats the existence of the natural number series as an hypothesis about the actual world ('axiom of infinity')."[20]

"The logicistic thesis can be questioned finally on the ground that logic already presupposes mathematical ideas in its formulation."[20]

Intuitionism

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The "belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384-322 B.C.), have an absolute validity, independent of the subject matter to which they are applied"[20] has been challenged.[25]

"According to his view and reading of history, classic logic was abstracted from the mathematics of finite sets and their subsets. ... Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification to the mathematics of infinite sets."[26]

Examples to show that principles "valid in thinking about finite sets do not necessarily carry over to infinite sets."[20]

  1. the whole is greater than any proper part, when applied to 1-1 correspondences between sets,
  2. a set of natural numbers contains a greatest.

Analogical reasoning

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Main resource: Analogical reasoning

Def. "a representational mapping from a known "source" domain into a novel "target" domain"[27] is called analogy.

"In problem solving and learning, analogical reasoning promises to overcome the explosive search complexity of finding solutions to novel problems or inducing generalized knowledge from experience."[27]

Def. "familiar [mapped] elements or relations from the source into unfamiliar (or unknown) elements or relations in the target" are called analogical inferences.[27]

"Source, target, mapping, analogical inference, and confirmatory support [a broad spectrum of empirical evidence] are the basic materials of analogy."[27]

Computer logic

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Main resource: Computer Logic

Computer logic is a system of principles behind the arrangements of elements in a computer or electronic device for performing a specified task.

Def. ordered "steps that solve a mathematical problem"[28] or a "precise step-by-step plan for a computational procedure that [possibly] begins with an input value and yields an output value in a finite number of steps"[29]

is called an algorithm.

Def. "an algorithm which calls itself with "smaller (or simpler)" input values, and which obtains the result for the current input by applying simple operations to the returned value for the smaller (or simpler) input"[30] is called a recursive algorithm.

Def. "a system that provides algorithms for the symbolic manipulation of first-order formulas over some temporarily fixed language and theory"[31] is called a computer logic system.

"The aim of logic in computer science is to develop languages to model the situations [encountered], in such a way that we can reason about them formally. Reasoning about situations means constructing arguments about them; we want to do this formally, so that the arguments are valid and can be defended rigorously, or executed on a machine."[32]

Debates

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Def.

  1. “a type of literary composition, taking the form of a discussion or disputation”[33]
  2. an "argument, or discussion, usually in an ordered or formal setting, often with more than two people, [generally] ending with a vote or other decision",[34]
  3. an "informal and spirited but generally civil discussion of opposing views",[35] and
  4. a discussion "of opposing views"[35]

is called a debate.

Deduction

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Main resource: Deduction

Def. a "process of reasoning that moves from the general to the specific, in which a conclusion follows necessarily from the premises presented, so that the conclusion cannot be false if the premises are true"[36] is called deduction.

Def. "inference in which the conclusion cannot be false given that the premises are true", or "Inference in which the conclusion is of no greater generality than the premises"[37] is called deductive reasoning.

Deductive reasoning, also called deductive logic, is the process of reasoning from one or more general statements regarding what is known to reach a logically certain conclusion.[38]

"The theory of deduction is intended to explain the relationship between premisses and conclusion of a valid argument and to provide techniques for the appraisal of deductive arguments"[17].

Dialectics

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Main resource: Dialectics

Dialectic (also dialectics and the dialectical method) is a method of argument for resolving disagreement. The dialectical method is dialogue between two or more people holding different points of view about a subject, who wish to establish the truth of the matter by dialogue, with reasoned arguments.[39] Dialectics is different from debate, wherein the debaters are committed to their points of view, and mean to win the debate, either by persuading the opponent, proving their argument correct, or proving the opponent's argument incorrect — thus, either a judge or a jury must decide who wins the debate. Dialectics is also different from rhetoric, wherein the speaker uses logos, pathos, or ethos to persuade listeners to take their side of the argument.

Induction

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Main resource: Induction

Def. "the derivation of general principles from specific instances"[40] or "a general proof of a theorem by first proving it for a specific integer (for example) and showing that, if it is true for one integer then it must be true for the next"[40] is called induction.

Inferences

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Main resource: Inferences

Inference is the act or process of deriving logical conclusions from premises known or assumed to be true.[41]

Logical calculus

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Main resource: Logical calculus

"[A]n abstract logical calculus [consists of] "the vocabulary of logic, ... the primitive symbols ..., and the logical structure ... fixed by stating the axioms or postulates ... in terms of its primitive symbols."[42]

Logic-based abduction

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Main resource: Logic-based abduction

In logic, explanation is done from a logical theory   representing a domain and a set of observations  . Abduction is the process of deriving a set of explanations of   according to   and picking out one of those explanations. To abduce   [  ∈  ] from   [  ∈  ] involves determining that   is sufficient (or nearly sufficient), but not necessary, for  .

"[T]o discover is simply to expedite an event that would occur sooner or later, if we had not troubled ourselves to make the discovery. Consequently, the art of discovery is purely a question of economics. The economics of research is, so far as logic is concerned, the leading doctrine with reference to the art of discovery. Consequently, the conduct of abduction, which is chiefly a question of heuretic and is the first question of heuretic, is to be governed by economical considerations."[43]

Mathematical logic

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Main resource: Mathematical logic

In line with Boolean algebra which is a logical calculus is Boolean logic.

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science.[44][45][46][47]

Natural deduction

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Main resource: Natural deduction

In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with the axiomatic systems which instead use axioms as much as possible to express the logical laws of deductive reasoning.

Principles

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Main resource: Principles

A principle is a law or rule that has to be, or usually is to be followed, or can be desirably followed, or is an inevitable consequence of something, such as the laws observed in nature or the way that a system is constructed. The principles of such a system are understood by its users as the essential characteristics of the system, or reflecting system's designed purpose, and the effective operation or use of which would be impossible if any one of the principles was to be ignored.[48]

Propositional logic

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Main resource: Propositional logic

Propositional logic uses or may result in declarative sentences.

Sophistry

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Main resource: Sophistry

Def. an "argument that seems plausible, but is fallacious or misleading, especially one devised deliberately to be so"[49] is called sophistry.

Symbolic logic

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Main resource: Symbolic logic

The systematic use of symbolic techniques to determine the forms of valid deductive argument may be deductive symbolic logic.

Trees

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"The logic tree is a pro/con hierarchy, in which the main debate topic is located at the root. Arguments are added in free text format and do not follow any specific structure. Users support (or attack) arguments by adding their own arguments under pro/yes (or con/no) sections of each question. Arguments can be re-used across debates; however, each debate tree is treated in isolation."[50]

Validity

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Main resource: Validity

Def. "the quality of state of [...] having a conclusion correctly derived from premises" is called validity.[4]

A sequent, e.g. ϕ₁, ϕ₂, ϕ₃, … ⊢ Ψ, is valid when a proof for it can be found[51].

An argument is a formula of the kind Premices → Conclusion and it is valid when for each interpretation under which the premises are all true, the conclusion is also true, or, in other words, when Premices ∧ ¬Conclusion = false.

This is also related with semantic entailment, e.g. ϕ₁, ϕ₂, ϕ₃, … ⊨ Ψ, which is a relation ⊨ that holds if Ψ evaluates to true whenever all formulas ϕ₁, ϕ₂, ϕ₃, … are evaluated to true.

Equivalently, a formula is defined as valid when it is true in every interpretation (is a tautology (logic)). To see this, it might be worth to rewrite ϕ₁, ϕ₂, ϕ₃, … ⊨ Ψ as its equivalent ⊨ ϕ₁∧ϕ₂∧ϕ₃∧… → Ψ.

A weaker concept, when formula can be true (but not necessary in all interpretations), is called satisfability. Valid formula is also satisfable but note vice-verse. However, negation relates the concepts more tightly: formula ϕ is satisfable iff ¬ϕ is not valid.

Hypotheses

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Main resource: Hypotheses
  1. Aristotelian logic is only a special case of validity-based logic.

See also

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References

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  1. ↑ Kompridis, Nikolas (2000). "So We Need Something Else for Reason to Mean". International Journal of Philosophical Studies 8 (3): 271–295. doi:10.1080/096725500750039282. 
  2. ↑ 2.0 2.1 individuals, for example, "humans have reason." Compare: MacIntyre, Alasdair (2013). Dependent Rational Animals: Why Human Beings Need the Virtues. The Paul Carus Lectures. Open Court. ISBN 978-0-8126-9705-6. https://books.google.com/books?id=Jv9Jx5iQ4uYC. Retrieved 2014-12-01. "[...] the exercise of independent practical reasoning is one essential constituent to full human flourishing." 
  3. ↑ Dan Polansky (6 November 2008). reasoning. San Francisco, California: Wikimedia Foundation, Inc. https://en.wiktionary.org/wiki/reasoning. Retrieved 2014-07-17. 
  4. ↑ 4.0 4.1 4.2 Noah Webster (1963). Philip Babcock Gove. ed. Webster's Seventh New Collegiate Dictionary. Springfield, Massachusetts: G. & C. Merriam Company. pp. 1221. https://books.google.com/books?id=r-QIAQAAIAAJ. Retrieved 2011-08-26. 
  5. ↑ Hintikka, Jaako. "Philosophy of logic". Encyclopædia Britannica. Encyclopædia Britannica, Inc. Retrieved 12 November 2013.
  6. ↑ Aristotle, Nicomachean Ethics 6 – The Intellectual Virtues
  7. ↑ Michel Foucault, "What is Enlightenment?" in The Essential Foucault, eds. Paul Rabinow and Nikolas Rose, New York: The New Press, 2003, 43–57. See also Nikolas Kompridis, "The Idea of a New Beginning: A Romantic Source of Normativity and Freedom," in Philosophical Romanticism, New York: Routledge, 2006, 32–59; "So We Need Something Else for Reason to Mean", International Journal of Philosophical Studies 8: 3, 271–295.
  8. ↑ Merriam-Webster.com Merriam-Webster Dictionary definition of reason
  9. ↑ Rachels, James. The Elements of Moral Philosophy, 4th ed. McGraw Hill, 2002
  10. ↑ Habermas, Jürgen (1990). The Philosophical Discourse of Modernity. Cambridge, MA: MIT Press. 
  11. ↑ Immanuel Kant, Critique of Pure Reason; Critique of Practical Reason.
  12. ↑ Michael Sandel, Justice: What's the Right Thing to Do?, New York: Farrar, Straus and Giroux, 2009.
  13. ↑ Kant, Immanuel; translated by James W. Ellington [1785] (1993). Grounding for the Metaphysics of Morals 3rd ed.. Hackett. p. 30. ISBN 978-0-87220-166-8. 
  14. ↑ 14.0 14.1 Jenny Teichmann and Katherine C. Evans, Philosophy: A Beginner's Guide (Blackwell Publishing, 1999), p. 1
  15. ↑ 15.0 15.1 A.C. Grayling (1998). Philosophy 1: A Guide through the Subject. Oxford University Press. pp. 1. 
  16. ↑ Anthony Quinton, in T. Honderich (ed.), The Oxford Companion to Philosophy (Oxford University Press, 1995), p. 666
  17. ↑ 17.0 17.1 17.2 17.3 Irving M. Copi (1955). Introduction to Logic. New York: The MacMillan Company. pp. 472. https://www.amazon.com/Introduction-Logic-Irving-M-Copi/dp/1292024828. Retrieved 2011-09-26. 
  18. ↑ "Historical Introduction to Philosophy/Philosophical Method". December 16, 2006. Retrieved 2011-11-29.
  19. ↑ "logic". San Francisco, California: Wikimedia Foundation, Inc. July 7, 2012. Retrieved 2012-07-30.
  20. ↑ 20.00 20.01 20.02 20.03 20.04 20.05 20.06 20.07 20.08 20.09 20.10 20.11 20.12 20.13 20.14 20.15 20.16 Stephen Cole Kleene (1971). Introduction to Metamathematics. Groningen: Wolters-Noordhoff Publishing. pp. 550. ISBN 0 444 10088 1. https://books.google.com/books/about/Introduction_to_metamathematics.html?id=GshaswEACAAJ. Retrieved 26 January 2020. 
  21. ↑ EncycloPetey (18 November 2006). "cardinal number". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 28 January 2020. {{cite web}}: |author= has generic name (help)
  22. ↑ NadandoBot (24 February 2019). "natural number". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 28 January 2020. {{cite web}}: |author= has generic name (help)
  23. ↑ Paul G (16 March 200). "natural number". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 28 January 2020. {{cite web}}: |author= has generic name (help)
  24. ↑ Alfred North Whitehead and Bertrand Russell (1927). Principia mathematica, 2nd edition. 1-3. Cambridge, England: University Press. ISBN 978-0521067911. https://www.amazon.com/Principia-Mathematica-Set-v-1-3/dp/052106791X. 
  25. ↑ L. E. J. Brouwer (1908). "De onbetrouwbaarheid der logische principes (The untrustworthiness of the principles of logic)". Tijdschrift voor wijsbegeerte 2: 152-8. 
  26. ↑ Hermann Weyl (1946). "Mathematics and logic. A brief survey serving as a preface to a review of The Philosophy of Bertrand Russell". The American Mathematical Monthly 53 (1): 2-13. doi:10.1080/00029890.1946.11991619. https://www.tandfonline.com/doi/abs/10.1080/00029890.1946.11991619?journalCode=uamm20. Retrieved 30 January 2020. 
  27. ↑ 27.0 27.1 27.2 27.3 Rogers P. Hall (May 1989). "Computational approaches to analogical reasoning: A comparative analysis". Artificial Intelligence 39 (1): 39-120. doi:10.1.1.94.7301. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.94.7301&rep=rep1&type=pdf. Retrieved 2012-07-30. 
  28. ↑ Niclisp (3 December 2015). algorithm. San Francisco, California: Wikimedia Foundation, Inc. https://en.wiktionary.org/wiki/algorithm. Retrieved 2017-08-11. 
  29. ↑ David M. Keil (2 September 2007). algorithm. San Francisco, California: Wikimedia Foundation, Inc. https://en.wiktionary.org/wiki/algorithm. Retrieved 2017-08-11. 
  30. ↑ Kenneth H. Rosen (Spring 2010). Recursive Algorithm. Norfolk, VA, USA: Old Dominion University. http://www.cs.odu.edu/~cs381/cs381content/recursive_alg/rec_alg.html. Retrieved 2017-08-11. 
  31. ↑ Andreas Dolzmann, Thomas Sturm (June 1997). "Redlog: Computer algebra meets computer logic". ACM SIGSAM Bulletin 31 (2): 2-9. doi:10.1145/261320.261324. http://dl.acm.org/citation.cfm?id=261320.261324. Retrieved 2012-07-30. 
  32. ↑ Michael Huth and Mark Ryan (August 26, 2004). Logic in Computer Science: Modelling and Reasoning about Systems. Cambridge, United Kingdom: Cambridge University Press. pp. 427. ISBN 0 521 54310 X. http://bilder.buecher.de/zusatz/12/12609/12609080_lese_1.pdf. Retrieved 2012-07-30. 
  33. ↑ Primetime~enwiktionary (25 February 2006). debate. San Francisco, California: Wikimedia Foundation, Inc. http://en.wiktionary.org/wiki/debate. Retrieved 2012-08-27. 
  34. ↑ Trunkie (28 February 2004). debate. San Francisco, California: Wikimedia Foundation, Inc. https://en.wiktionary.org/wiki/debate. Retrieved 2017-08-11. 
  35. ↑ 35.0 35.1 Dmh (28 February 2004). debate. San Francisco, California: Wikimedia Foundation, Inc. https://en.wiktionary.org/wiki/debate. Retrieved 2017-08-11. 
  36. ↑ deduction. San Francisco, California: Wikimedia Foundation, Inc. April 11, 2012. http://en.wiktionary.org/wiki/deduction. Retrieved 2012-07-30. 
  37. ↑ deductive reasoning. San Francisco, California: Wikimedia Foundation, Inc. May 21, 2012. http://en.wiktionary.org/wiki/deductive_reasoning. Retrieved 2012-07-30. 
  38. ↑ R. J. Sternberg (2009). Cognitive Psychology. Belmont, CA: Wadsworth. pp. 578. ISBN 978-0-495-50629-4. 
  39. ↑ The Republic (Plato), 348b
  40. ↑ 40.0 40.1 SemperBlotto (14 July 2005). induction. San Francisco, California: Wikimedia Foundation, Inc. https://en.wiktionary.org/wiki/induction. Retrieved 2018-03-11. 
  41. ↑ http://www.thefreedictionary.com/inference
  42. ↑ Patrick Suppes (1967). Sidney Morgenbesser. ed. What is a scientific theory? In: Philosophy of Science Today. New York: Basic Books, Inc.. pp. 55-67. http://suppescorpus.stanford.edu/articles/mpm/84.pdf. Retrieved 2011-12-15. 
  43. ↑ Peirce, C.S. (1902), application to the Carnegie Institution, see MS L75.329-330, from Draft D of Memoir 27
  44. ↑ Boolos, George; Burgess, John; Jeffrey, Richard (2002). Computability and Logic (4th ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-00758-0. 
  45. ↑ Enderton, Herbert (2001). A mathematical introduction to logic (2nd ed.). Boston, MA: Academic Press. ISBN 978-0-12-238452-3. 
  46. ↑ Mendelson, Elliott (1997). Introduction to Mathematical Logic (4th ed.). London: Chapman & Hall. ISBN 978-0-412-80830-2. 
  47. ↑ Shoenfield, Joseph R. (2001) [1967]. Mathematical Logic (2nd ed.). A K Peters. ISBN 978-1-56881-135-2. 
  48. ↑ Alpa, Guido (1994). "General Principles of Law". Annual Survey of International & Comparative Law 1 (1, Article 2). http://digitalcommons.law.ggu.edu/annIsurvey/voll/issl/2. 
  49. ↑ sophistry. San Francisco, California: Wikimedia Foundation, Inc. 19 June 2014. https://en.wiktionary.org/wiki/sophistry. Retrieved 2014-07-17. 
  50. ↑ Iyad Rahwan, Bita Banihashemi, Chris Reed, Douglas Walton, and Sherief Abdallah (December 2011). "Representing and classifying arguments on the semantic web". The Knowledge Engineering Review 26 (4): 487-511. http://www.academia.edu/download/45633671/Representing_and_classifying_arguments_o20160514-24886-xgrixj.pdf. Retrieved 2017-08-11. 
  51. ↑ Logic in Computer Science: Modelling and Reasoning

Further reading

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{{Semantics resources}}

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