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Broadly speaking, a contradiction is an incompatibility between two or more statements, ideas, or actions. One must, it seems, reject at least one of the ideas outright.

In logic, contradiction is defined much more specifically, usually as the simultaneous assertion of a statement and its negation ("denial" can be used instead of "negation"). This, of course, assumes that "negation" has a non-problematic definition. This idea is based on Aristotle's law of non-contradiction which states that "One cannot say of something that it is and that it is not in the same respect and at the same time." Logic, from Classical Greek Î»ÏŒÎ³Î¿Ï‚ (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... Negation (i. ... Aristotle (Ancient Greek: AristotelÄ“s 384 BC â€“ March 7, 322 BC) was an ancient Greek philosopher, who studied with Plato and taught Alexander the Great. ... In logic, the law of noncontradiction judges as false any proposition P asserting that both proposition Q and its denial, proposition not-Q, are true at the same time and in the same respect. In the words of Aristotle, One cannot say of something that it is and that it...

In TRIZ contradiction (as one of the basic definitions) is a situation where an attempt to improve one feature of the system causes deterioration of another feature (for example: If we want more acceleration, we need a larger engine - but that will increase the cost of the car). TRIZ, (pronounced [triz]), is a Russian acronym for Teoriya Resheniya Izobretatelskikh Zadatch (Ð¢ÐµÐ¾Ñ€Ð¸Ñ Ñ€ÐµÑˆÐµÐ½Ð¸Ñ Ð¸Ð·Ð¾Ð±Ñ€ÐµÑ‚Ð°Ñ‚ÐµÐ»ÑŒÑÐºÐ¸Ñ… Ð·Ð°Ð´Ð°Ñ‡), a Theory of solving inventive problems or Theory of inventive problems solving (TIPS) (less known as Theory of Solving Inventors Problems), developed by Genrich Altshuller and his colleagues since 1946. ... A definition delimits or describes the meaning of a concept or term by stating the essential properties of the entities or objects denoted by that concept or term. ... In geographic information systems, a feature comprises an item of feature data. ... Look up system in Wiktionary, the free dictionary For the Macintosh operating system, which was called System up to version 7. ...

In colloquial speech and in dialectical methodology, the word "contradiction" has a completely different meaning than in formal logic.

### In colloquial speech

In everyday speech, "contradiction" may be used in a much less rigorous way than in formal logic. For example, there is nothing logically contradictory involved in a man condemning the members of his church for not giving the church enough financial support even though he never puts anything in the collection plate when it goes around. In ordinary language we would be quite inclined to say that his actions contradict his words, but the immediate connection of this usage to the logical usage is unclear. Hypocrisy is certainly lamentable but it's hard to say that it's logically incoherent--our hypothetical church-goer, after all, is not clearly asserting anything by refusing to put money in the collection plate, let alone the logical negation of what he asserted.

One way to understand the colloquial usage might be to shift grounds from logical contradiction to what some philosophers describe as a performative contradiction. A hypocrite is not saying anything that contradicts the general principles that he asserts to be true; but his actions, in some sense, presuppose that those principles are false. Similarly, "I cannot assert anything." is a sentence that no-one can truly utter. This is not because of a logical contradiction in the sentence--it is, for example, true of the brain-dead. But there is a performative contradiction involved in the act of saying it; for to say it presupposes that you can assert something. In pragmatics, a presupposition is an assumption about the world whose truth is taken for granted in discourse. ...

### In Dialectics

#### Marxism

The meaning of "contradiction" in dialectical materialism pertains to the views of G.W.F. Hegel and Karl Marx. In this usage a "contradiction" does not refer to a conflict purely in a person's thinking or in logic, but indicates a clash between one's theory and one's practice, or one's words and one's deeds. It has been suggested that Marxist philosophy of nature be merged into this article or section. ... Georg Wilhelm Friedrich Hegel (August 27, 1770 - November 14, 1831) was a German philosopher born in Stuttgart, Württemberg, in present-day southwest Germany. ... Karl Heinrich Marx (May 5, 1818 Trier, Germany â€“ March 14, 1883 London) was an immensely influential German philosopher, political economist, and revolutionary organizer of the International Workingmens Association. ...

This meaning of contradiction is more of a practical, empirical, or real-world phenomenon than is meant by a logic-based contradiction. For Marx, capitalism involves a social system that has "contradictions" in the sense that the social classes have conflicting collective goals, and in the sense that even the ruling class of capitalists does not always attain their goals. In Marx's view, real-world contradictions are based in the social structure of the society in question, and inherently lead to class conflict, crisis, and eventually revolution where the existing order is overthrown and the formerly oppressed class rises up and assumes political power. Capitalism is commonly understood to mean an economic or socioeconomic system in which the means of production are predominantly privately owned and operated for profit, often through the employment of labour. ... Social class refers to the hierarchical distinctions between individuals or groups in societies or cultures. ... In Marxian political economics, the ruling class refers to that segment or class of society that has the most economic and political power. ... In economics, a capitalist is someone who owns capital, presumably within the economic system of capitalism. ... Class conflict is both the friction that accompanies social relationships between members or groups of different social classes and the underlying tensions or antagonisms which exist in society. ... In economics, crisis is an old term in business cycle theory, referring to the sharp transition to a recession. ... It has been suggested that Revolutionary be merged into this article or section. ...

#### Liberalism

The idea of a contradiction as a conflict based in a social structure is not unique to Marxist thought. For liberal thinkers, the problem of public goods may be interpreted as a "contradiction" in that there is a conflict between what is good for society, i.e., the production of a public good, and what is good for individual free riders who refuse to pay the costs of the public good. This is one interpretation of Hegel's view of contradictions, seen for example in Paul Deising, Hegel's Dialectical Political Economy (ISBN 0813391318). Marxism is the political practice and social theory based on the works of Karl Marx, a 19th century philosopher, economist, journalist, and revolutionary, along with Friedrich Engels. ... Look up liberal on Wiktionary, the free dictionary Liberal may refer to: Politics: Liberalism American liberalism, a political trend in the USA Political progressivism, a political ideology that is for change, often associated with liberal movements Liberty, the condition of being free from control or restrictions Liberal Party, members of... In economics, a public good is one that cannot or will not be produced for individual profit, since it is difficult to get people to pay for its large beneficial externalities. ... In economics and political science, free riders are actors who consume more than their fair share of a resource, or shoulder less than a fair share of the costs of its production. ...

In deductive logic (and thus, also, in mathematics), a contradiction is usually taken as a sign that something has gone wrong, that you need to retrace the steps of your reasoning and "check your premises." This has been used to great effect in mathematics through the method of proof by contradiction: since a contradiction can never be true, it can thus never be the conclusion of a valid argument with all true premises. To construct a proof by contradiction, then, you construct a valid proof from a set of premises to a conclusion that is a logical contradiction. Since the conclusion is false, and the argument is valid, the only possibility is that one or more of the premises are false. This method is used in many key mathematical proofs, such as Euclid's proof that there is no greatest prime, and Cantor's diagonal proof that there are uncountably many real numbers between 0 and 1. Euclid, detail from The School of Athens by Raphael. ... Reductio ad absurdum (Latin for reduction to the absurd, traceable back to the Greek ἡ εις το αδυνατον απαγωγη, reduction to the impossible, often used by Aristotle) is a type of logical argument where we assume a claim for the sake of argument, arrive at an absurd result, and then... Note: in order to fully understand this article you may want to refer to the set theory portion of the table of mathematical symbols. ...

Contradiction is associated with several notorious paradoxes. One of these is that in first-order predicate calculus any proposition (aka statement) can be derived from a contradiction. In other words, according to the predicate calculus, no matter what P and Q mean, if P and not-P are both true, then Q is true. In expression of this fact, contradictions are said to be "logically explosive" in first-order logic. First-order predicate calculus or first-order logic (FOL) permits the formulation of quantified statements such as there exists an x such that. ... Proposition is a term used in logic to describe the content of assertions. ...

Thus, for example, the following argument is strictly valid, i.e. the premise logically entails the conclusion: This article discusses validity in logic, for the term in the social sciences see validity (psychometric). ...

1. Premise: 5 is both even and odd. (In our above formulation, this is P and not-P.)
2. Conclusion: God exists. (This is Q.)

But atheists have no less reason to celebrate than theists, for this argument is also valid: In mathematics, the parity of an object refers to whether is is even or odd. ... In mathematics, the parity of an object refers to whether is is even or odd. ...

1. Premise: 5 is both even and odd. (This is P and not-P.)
2. Conclusion, God does not exist. (This is not-Q.)

Note that the premise shared by both arguments is incorrect; 5 is odd, but is not even. Therefore neither of these arguments are sound, which means neither gives a logical basis for believing its conclusion. (This article discusses the soundess notion of informal logic. ...

Nonetheless, perhaps most people find it odd that, if 5 were both even and odd, one could logically conclude anything about such an apparently unrelated matter as the existence of God. Stranger yet, the paradox implies that, if a person has any two beliefs that are contradictory, then that person is logically justified in any conceivable belief!

Even though the basic rules of predicate calculus may each sound like good ways of reasoning, they collectively entail our paradox. Two ways of showing this follow.

The first way follows from the truth table definition of conjunction and implication:

1. (P and ¬P) is false. (See the truth table entry for PQ.)
2. Therefore, (P and ¬P) → Q is vacuously true. (See the truth table entry for PQ.)

The second then, might interest those who find truth tables aesthetically flawed: Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ... Informally, a logical statement is vacuously true if it is true but doesnt say anything; examples are statements of the form everything with property A also has property B, where there is nothing with property A. It is tempting to dismiss this concept as vacuous or silly. ... Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

1. Suppose P and ¬P. Under this assumption we can derive:
1. P (Conjunction elimination)
2. ¬P (Conjunction elimination)
3. Suppose ¬Q. Under this assumption we can derive:
1. P (Copying from above)
4. Thus ¬Q → P (Conditional proof)
5. ¬P → Q (Contrapositive of previous line)
6. Q (Modus ponens)
2. Thus (P and ¬P) → Q (Conditional proof)

In logic, conjunction elimination is the inference that, if the conjunction A and B is true, then A is true, and B is true. ... In logic, conjunction elimination is the inference that, if the conjunction A and B is true, then A is true, and B is true. ... Conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion. ... In predicate logic, the contrapositive (or transposition) of the statement p implies q is not-q implies not-p. ... In Logic, Modus ponens (Latin: mode that affirms) is a valid, simple argument form (often abbreviated to MP): If P, then Q. P. Therefore, Q. or in logical operator notation: P â†’ Q P âŠ¢ Q where âŠ¢ represents the logical assertion. ... Conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion. ...

Coherentism is an epistemological theory in which a belief is justified based at least in part on being part of a non-contradictory system of beliefs. ("Contradictory" here is almost always taken in the formal logic sense.) Coherentism - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... Epistemology is an analytic branch of philosophy which studies the nature, origin, and scope of knowledge. ...

It often occurs in philosophy that the presence of the argument contradicts with the claims of the argument. An example of this is when Heraclitus says that knowledge is impossible, or arguably when Nietzsche says that you should not obey others, you should not be obeying his statement. There are many similar examples. Often coherentism is temporarily ignored for these theories, and a relief is granted to the philosophers because there is no other way to explain the theory. Heraclitus by Johannes Moreelse Heraclitus of Ephesus (Greek Herakleitos) (about 535 - 475 BC), known as The Obscure, was a pre-Socratic Greek philosopher from Ephesus in Asia Minor. ... Friedrich Nietzsche, 1882 Friedrich Wilhelm Nietzsche (October 15, 1844 - August 25, 1900) was a highly influential German philosopher. ... Coherentism - Wikipedia, the free encyclopedia /**/ @import /skins-1. ...

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