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| :''This article uses forms of [[Mathematical logic|logical]] notation. For a concise description of the symbols used in this notation, see [[List of logic symbols]].''
| | Of all the things that this company ought to aware of, then it is certainly with variability. Set it simply, for a businessman in Australia, variability means opportunities. Indeed, if one is to have a look in the current Australian market, additionally might choose to the conclusion that the united states has lots of hours of potential associated with online business venture. To support it, it is a good idea to choose hosting solutions that can maximise a website's performance. To that end, investing in cloud hosts would be a may. There are associated with reasons why this certainly.<br><br>Ergonomics could be the study of how an item connects to a person and also how it affects the persons health. Searching an ergonomic furnishings site can offer solutions within your pain problems in a chair couple of back support or a wrist rest for your keyboard other individuals. These items might be more costly but technique prevent pain as you are working.<br><br><br><br>There is technology that permits hydrogen pertaining to being removed from water and burnt as fuel. Program that runs your car on water is easily installed in fact it is my first choice to better the environment. It is a clean burning readily available alternative.<br><br>You a lot more have for limited to be able to a few colors. May do choose from an choice of colors regarding example the Pink Curve 8520 and the White Curve 8520. This means women are now able to choose one that very good more at ease. If you prefer a cleaner look, you can choose the phone's white transcription.<br><br>Hopefully, while much with a lot of other areas regarding home improvement, you'll then need shell out more focus to some things than while others. Do take a detailed look at what you need, and after that make a determination concerning how very different things sign up to you. Because realize, hard work even more to account than just what available below. The balance associated with this read contains much more that may well your specific situation. We know you will find them strongly related your overall goals, plus there is even far.<br><br>This System works using a 5 pip trailing stop and a 20 pip take earn money. If the signal falls between 5 pips and 20 pips, Forex Ambush 2.0 gives a trade connection. If it falls aside from this range, Forex Ambush 2.0 advises you in order to trade. Machine does not use hard stop loss technique.<br><br>Judge and judge some more. You judge a products or services by comparing it to competing or similar goods. You can also attempt to benchmark make use of this against its own matters.<br><br>Bioethanols are pretty much the gasoline's equivalent to biodiesel. Those are the most used often biofuel the particular world, and are usually pretty darn expensive. Learn how to run your own vehicle on water and save lots of money with the particular amount placed.<br><br>If you have any inquiries about where and how to use [http://www.ractiv.com/ 3D], you can call us at our own web page. |
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| In [[classical logic]], the '''law of non-contradiction''' (LNC) (or the '''law of contradiction''' (PM) or the '''principle of non-contradiction''' (PNC), or the '''principle of contradiction''') is the second of the [[three classic laws of thought]]. It states that contradictory statements cannot both be true in the same sense at the same time, e.g. the two propositions "''A is B''" and "''A is not B''" are mutually exclusive.
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| The principle was stated as a [[theorem]] of [[propositional calculus|propositional logic]] by [[Bertrand Russell|Russell]] and [[Alfred North Whitehead|Whitehead]] in ''[[Principia Mathematica]]'' as:
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| :: <math>\mathbf{*3\cdot24}. \ \ \vdash. \thicksim(p.\thicksim p)</math><ref>{{citation|author=[[Alfred North Whitehead]], [[Bertrand Russell]]|title=Principia Mathematica|publisher=[[Cambridge]]|year=1910|pages=116–117}}[http://name.umdl.umich.edu/aat3201.0001.001]</ref>
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| The law of noncontradiction, along with its complement, the [[law of excluded middle]] (the third of the three classic laws of thought), are correlates of the [[law of identity]] (the first of the three laws). Because the law of identity partitions its logical Universe into exactly two parts, it creates a [[dichotomy]] wherein the two parts are "mutually exclusive" and "jointly exhaustive". The law of noncontradiction is merely an expression of the mutually exclusive aspect of that dichotomy, and the law of excluded middle, an expression of its jointly exhaustive aspect.
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| == Interpretations ==
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| One difficulty in applying the law of noncontradiction is ambiguity in the propositions. For instance, if time is not explicitly specified as part of the propositions A and B, then ''A'' may be ''B'' at one time, and not at another. A and B may in some cases be made to sound mutually exclusive linguistically even though ''A'' may be partly ''B'' and partly not ''B'' at the same time. However, it is impossible to predicate of the same thing, at the same time, and in the same sense, the absence and the presence of the same fixed quality.
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| === Eastern philosophy ===
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| The law of noncontradiction is found in ancient [[Indian logic]] as a meta-rule in the ''[[Kalpa (Vedanga)|Shrauta Sutras]]'', the grammar of [[Pāṇini]],<ref>{{citation|author=[[Frits Staal]]|title=Universals: Studies in Indian Logic and Linguistics|publisher=[[Chicago]]|year=1988|pages=109–28}} ([[cf.]] {{citation|title=Seeing Things Hidden|first=Malcolm|last=Bull|publisher=Verso|year=1999|isbn=1-85984-263-1|page=53}})</ref> and the ''[[Brahma Sutras]]'' attributed to [[Vyasa]]. It was later elaborated on by medieval commentators such as [[Madhvacharya]].<ref>{{citation|title=A History of Indian Philosophy|first=Surendranath|last=Dasgupta|publisher=[[Motilal Banarsidass]]|year=1991|isbn=81-208-0415-5|page=110}}</ref>
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| === Heraclitus ===
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| According to both Plato and Aristotle,<ref>{{citation|title=Metaphysics (IV,1005b)|last=Aristotle |quote=to suppose that the same thing is and is not, as some imagine that Heraclitus says}}</ref> [[Heraclitus]] was ''said'' to have denied the law of noncontradiction. This is quite likely<ref>{{citation|title=Fragments 36,57,59 (Bywater)|last=Heraclitus}}</ref> if, as Plato [[Law of Contradiction#Plato's synthesis|pointed out]], the law of noncontradiction does not hold for changing things in the world. If a philosophy of [[Becoming (philosophy)|Becoming]] is not possible without change, then (the potential of) what is to become must already exist in the present object. In "''We step and do not step into the same rivers; we are and we are not''", both Heraclitus's and Plato's object simultaneously must, in some sense, be both what it now is and have the potential (dynamis) of what it might become.<ref>{{citation| last=Cornford| first=F.M.| title=Plato's Theory of Knowledge|page=234}}</ref>
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| Unfortunately, so little remains of [[Heraclitus]]' aphorisms that not much about his philosophy can be said with certainty. He seems to have held that strife of opposites is universal both within and without, therefore ''both'' opposite existents or qualities must simultaneously exist, although in some instances in different respects. "The ''road up and down are one and the same''" implies either the road leads both ways, or there can be no road at all. This is the logical [[Complement (set theory)|complement]] of the law of noncontradiction. According to [[Heraclitus]], change, and the constant conflict of opposites is the universal [[logos]] of nature.
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| === Protagoras ===
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| Personal subjective perceptions or judgments can only be said to be true at the same time in the same respect, in which case, the law of noncontradiction must be applicable to personal judgments.
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| The most famous saying of [[Protagoras]] is: "''Man is the measure of all things: of things which are, that they are, and of things which are not, that they are not''".<ref>(80B1 [[Diels-Kranz|DK]]). According to Plato's ''[[Theaetetus (dialogue)|Theaetetus]]'', section 152a. [http://www.perseus.tufts.edu/cgi-bin/ptext?lookup=Plat.+Theaet.+152a]</ref> However, Protagoras was referring to things that are used by or in some way related to humans. This makes a great difference in the meaning of his aphorism. Properties, social entities, ideas, feelings, judgements, etc. originate in the human mind. However, Protagoras has never suggested that man must be the measure of stars, or the motion of the stars.
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| === Parmenides ===
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| [[Parmenides]] employed an [[ontological]] version of the law of noncontradiction to prove that being is and to deny the void, change, and motion. He also similarly disproved contrary propositions. In his poem [[On Nature]], he said,
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| {{Quotation|the only routes of inquiry there are for thinking:<br>
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| the one that [it] is and that [it] cannot not be<br>
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| is the path of Persuasion (for it attends upon truth)<br>
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| the other, that [it] is not and that it is right that [it] not be,<br>
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| this I point out to you is a path wholly inscrutable<br>
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| for you could not know what is not (for it is not to be accomplished)<br>
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| nor could you point it out… For the same thing is for thinking and for being<br>}}
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| The nature of the ‘is’ or what-is in Parmenides is a highly contentious subject. Some have taken it to be whatever exists, some to be whatever is or can be the object of scientific inquiry.<ref>Curd, Patricia, "Presocratic Philosophy", ''The Stanford Encyclopedia of Philosophy (Summer 2011 Edition)'', Edward N. Zalta (ed.), URL = http://plato.stanford.edu/archives/sum2011/entries/presocratics/</ref>
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| === Socrates ===
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| In Plato's early dialogues, Socrates uses the [[Socratic method#Method|elenctic method]] to investigate the nature or definition of ethical concepts such as justice or virtue. Elenctic refutation depends on a [[dichotomy|dichotomous]] thesis, one that may be divided into exactly two [[mutually exclusive]] parts, only one of which may be true. Then Socrates goes on to demonstrate the contrary of the commonly accepted part using the law of noncontradiction. According to Gregory Vlastos,<ref>Gregory Vlastos, 'The Socratic Elenchus', ''Oxford Studies in Ancient Philosophy I'', Oxford 1983, 27–58.</ref> the method has the following steps:
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| # Socrates' [[Interlocutor (linguistics)|interlocutor]] asserts a thesis, for example "Courage is endurance of the soul", which Socrates considers false and targets for refutation.
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| # Socrates secures his interlocutor's agreement to further premises, for example "Courage is a fine thing" and "Ignorant endurance is not a fine thing".
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| # Socrates then argues, and the interlocutor agrees, that these further premises imply the contrary of the original thesis, in this case it leads to: "courage is not endurance of the soul".
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| # Socrates then claims that he has shown that his interlocutor's thesis is false and that its negation is true.
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| === Plato's synthesis ===
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| [[Plato]]'s version of the law of noncontradiction states that "''The same thing clearly cannot act or be acted upon in the same part or in relation to the same thing at the same time, in contrary ways''" (The ''[[Republic (dialogue)|Republic]]'' (436b)). In this, Plato carefully phrases three [[axiom]]atic restrictions on ''action'' or reaction: 1) in the same part, 2) in the same relation, 3) at the same time. The effect is to momentarily create a frozen, timeless [[state of affairs (philosophy)|state]], somewhat like figures frozen in action on the frieze of the Parthenon.<ref>[http://www.the-philosopher.co.uk/lawsofthought.htm James Danaher, ''The Laws of Thought''] "The restrictions Plato places on the laws of thought (i.e., "in the same respect," and "at the same time,") are an attempt to isolate the object of thought by removing it from all other time but the present and all respects but one."</ref>
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| This way, he accomplishes two essential goals for his philosophy. First, he logically separates the Platonic world of constant change<ref>Plato's [[Analogy of the divided line|Divided Line]] describes the four Platonic worlds</ref> from the formally knowable world of momentarily fixed physical objects.<ref>''Cratylus'', starting at [http://www.perseus.tufts.edu/hopper/text?doc=Plat.+Crat.+439e&fromdoc=Perseus%3Atext%3A1999.01.0172 439e]</ref><ref>"A thing which is F at one time, or in one way, or in one relation, or from one point of view, will be all too often not-F, at another time, in another way" ("Metaphysical Paradox" in Gregory Vlastos, ''Platonic Studies'', p.50)</ref> Second, he provides the conditions for the [[dialectic]] method to be used in finding definitions, as for example in the ''[[Sophist (dialogue)|Sophist]]''. So Plato's law of noncontradiction is the empirically derived necessary starting point for all else he has to say.
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| In contrast, Aristotle reverses Plato's order of derivation. Rather than starting with ''experience'', Aristotle begins ''a priori'' with the law of noncontradiction as the fundamental axiom of an analytic philosophical system.<ref>Similarly, Kant remarked that Newton "''by no means dared to prove this law a priori, and therefore appealed rather to experience''" (''Metaphysical Foundations'', 4:449)</ref> This axiom then necessitates the fixed, realist model. Now, he starts with much stronger logical foundations than Plato's non-contrariety of action in reaction to conflicting demands from the three parts of the soul.
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| === Aristotle's contribution ===
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| The traditional source of the law of noncontradiction is [[Aristotle]]'s ''[[Metaphysics (Aristotle)|Metaphysics]]'' where he gives three different versions.<ref>{{Harvtxt|Łukasiewicz|1971}} p.487</ref>
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| #[[ontological]]: "It is impossible that the same thing belong and not belong to the same thing at the same time and in the same respect." (1005b19-20)
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| #[[psychological]]: "No one can believe that the same thing can (at the same time) be and not be." (1005b23-24)
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| #[[logical]]: "The most certain of all basic principles is that contradictory [[proposition]]s are not true simultaneously." (1011b13-14)
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| Aristotle attempts several proofs of this law. He first argues that every expression has a single meaning (otherwise we could not communicate with one another). This rules out the possibility that by "to be a man", "not to be a man" is meant. But "man" means "two-footed animal" (for example), and so if anything is a man, it is necessary (by virtue of the meaning of "man") that it must be a two-footed animal, and so it is impossible at the same time for it ''not'' to be a two-footed animal. Thus "it is not possible to say truly at the same time that the same thing is and is not a man" (''Metaphysics'' 1006b 35). Another argument is that anyone who believes something cannot believe its contradiction (1008b).
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| :Why does he not just get up first thing and walk into a well or, if he finds one, over a cliff? In fact, he seems rather careful about cliffs and wells.<ref>1008b, trans. Lawson-Tancred</ref>
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| [[Avicenna]] gives a similar argument:
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| :Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned.<ref>Avicenna, Metaphysics, I; commenting on Aristotle, Topics I.11.105a4–5.</ref><ref>Of course, if Avicenna was burned on the one hand while frozen on the other, he would have soon admitted that it is not the same to be ''both'' burned and frozen or ''neither''.</ref>
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| === Leibniz and Kant ===
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| [[Gottfried Wilhelm Leibniz|Leibniz]] and [[Immanuel Kant|Kant]] adopted a different statement, by which the law assumes an essentially different meaning. Their formula is A is not not-A; in other words it is impossible to predicate of a thing a quality which is its contradictory. Unlike Aristotle's law this law deals with the necessary relation between subject and predicate in a single judgment. For example, in [[Gottlob Ernst Schulze]]'s ''[[Aenesidemus (book)|Aenesidemus]]'', it is asserted, "… nothing supposed capable of being thought may contain contradictory characteristics." Whereas Aristotle states that one or other of two contradictory propositions must be false, the Kantian law states that a particular kind of proposition is in itself necessarily false. On the other hand there is a real connection between the two laws. The denial of the statement A is not-A presupposes some knowledge of what A is, i.e. the statement A is A. In other words a judgment about A is implied.
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| Kant's analytical judgments of propositions depend on presupposed concepts which are the same for all people. His statement, regarded as a logical principle purely and apart from material facts, does not therefore amount to more than that of Aristotle, which deals simply with the significance of negation{{Citation needed|date=April 2009}}.
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| === Modern logics ===
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| Traditionally, in [[Aristotle]]'s classical [[Term logic|logical calculus]], in evaluating any [[proposition]] there are only two possible [[truth value]]s, "true" and "false." An obvious extension to classical two-valued logic is a many-valued logic for more than two possible values. In [[logic]], a many- or multi-valued logic is a [[propositional calculus]] in which there are more than two values. Those most popular in the literature are [[Three-valued logic|three-valued]] (e.g., [[Jan Łukasiewicz|Łukasiewicz's]] and [[Stephen Cole Kleene|Kleene's]]), which accept the values "true", "false", and "unknown", finite-valued with more than three values, and the infinite-valued (e.g. [[fuzzy logic]] and [[probabilistic logic|probability logic]]) logics.
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| === Dialetheism ===
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| [[Graham Priest]] advocates the view that ''under some conditions'', some statements can be both true and false simultaneously, or may be true and false at different times. Applied universally, without specified conditions or [[axiom]]atic restrictions, this [[dialetheism]] will cause every statement, to [[principle of explosion|explode]], to become true. Dialetheism arises from formal logical [[paradox]]es, such as the [[liar paradox|Liar's paradox]] and [[Russel's paradox|Russell's paradox]].
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| == Alleged impossibility of its proof or denial ==
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| As is true of all [[axioms]] of logic, the law of non-contradiction is alleged to be neither verifiable nor falsifiable, on the grounds that any proof or disproof must use the law itself prior to reaching the conclusion. In other words, in order to verify or falsify the laws of logic one must resort to logic as a weapon, an act which would essentially be [[Self-refuting idea|self-defeating]].<ref>[http://faculty.washington.edu/smcohen/AristotlePNC.pdf S.M. Cohen, ''Aristotle on the Principle of Non-Contradiction''] "''Aristotle's solution in the Posterior Analytics is to distinguish between episteme (scientific knowledge) and nous (intuitive intellect). First principles, such as PNC, are not objects of scientific knowledge - since they are not demonstrable - but are still known, since they are grasped by nous''".</ref> Since the early 20th century, certain logicians have proposed logics that deny the validity of the law. Collectively, these logics are known as "[[paraconsistent logic|paraconsistent]]" or "inconsistency-tolerant" logics. But not all paraconsistent logics deny the law, since they are not necessarily completely agnostic to inconsistencies in general. [[Graham Priest]] advances the strongest thesis of this sort, which he calls "[[dialetheism]]".
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| In several axiomatic derivations of logic,<ref>Steven Wolfram, A New Kind Of Science, ISBN 1-57955-008-8</ref> this is effectively resolved by showing that (P ∨ ¬P) and its negation are constants, and simply defining TRUE as (P ∨ ¬P) and FALSE as ¬(P ∨ ¬P), without taking a position as to the [[principle of bivalence]] or the [[law of excluded middle]].
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| Some, such as [[David Lewis (philosopher)|David Lewis]], have objected to paraconsistent logic on the ground that it is simply impossible for a statement and its negation to be jointly true.<ref>See Lewis (1982).</ref> A related objection is that "negation" in paraconsistent logic is not really ''[[negation]]''; it is merely a [[Square of opposition|subcontrary]]-forming operator.<ref>See Slater (1995), Béziau (2000).</ref>
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| == See also ==
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| * [[Contradiction]]
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| * [[First principle]]
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| * [[Identity (philosophy)]]
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| * [[Law of excluded middle]]
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| * [[Law of identity]]
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| * [[Laws of thought]]
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| * [[Liar's Paradox]]
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| * [[Peirce's law]]
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| * [[Principle of bivalence]]
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| * [[Principle of explosion]]
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| * [[Reductio ad absurdum]]
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| * [[Three classic laws of thought]]
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| * [[Oxymoron]]
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| == Notes ==
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| <references/>
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| == References ==
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| *{{cite book|title=Socrates' Second Sailing: On Plato’s Republic|last=Benardete|first=Seth|year=1989|publisher=University of Chicago Press}}
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| *{{cite book|title=Aristotle's Metaphysics|last=Lawson-Tancred|first=H|year=1998|publisher=Penguin}}
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| *{{Citation|last=Łukasiewicz|first=Jan|year=1971|title=On the Principle of Contradiction in Aristotle|journal=Review of Metaphysics|volume=24|pages=485–509|year=1910 (in Polish)}}
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| == External links ==
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| * S.M. Cohen, ''"[http://faculty.washington.edu/smcohen/AristotlePNC.pdf Aristotle on the Principle of Non-Contradiction]"'', ''Canadian Journal of Philosophy'', Vol. 16, No. 3
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| * James Danaher, ''"[http://www.the-philosopher.co.uk/lawsofthought.htm The Laws of Thought]"'', ''The Philosopher'', Vol. LXXXXII No. 1
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| * Paula Gottlieb, "[http://plato.stanford.edu/entries/aristotle-noncontradiction/ Aristotle on Non-contradiction]" ([[Stanford Encyclopedia of Philosophy]])
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| * [[Laurence Horn]], "[http://plato.stanford.edu/entries/contradiction/ Contradiction]" ([[Stanford Encyclopedia of Philosophy]])
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| * [[Graham Priest]] and Francesco Berto, "[http://plato.stanford.edu/entries/dialetheism/ Dialetheism]" ([[Stanford Encyclopedia of Philosophy]])
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| * Graham Priest and Koji Tanaka, "[http://plato.stanford.edu/entries/logic-paraconsistent/ Paraconsistent logic]" ([[Stanford Encyclopedia of Philosophy]])
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| * Peter Suber, ''"[http://www.earlham.edu/~peters/courses/logsys/pnc-pem.htm Non-Contradiction and Excluded Middle]"'', Earlham College
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| [[Category:Classical logic]]
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| [[Category:Theorems in propositional logic]]
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