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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." In TRIZ contradiction (as one of the basic Definition s) 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''). In colloquial speech and in dialectical methodology, the word "contradiction" has a completely different meaning than in formal logic. "CONTRADICTION" OUTSIDE OF 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 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. 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 Class es have conflicting collective goals, and in the sense that even the Ruling Class of Capitalist s 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. 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). "CONTRADICTION" IN FORMAL LOGIC Proof by contradiction In deductive logic (and thus, also, in that there are uncountably many real numbers between 0 and 1. A paradox involving contradiction 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. Thus, for example, the following argument is ''strictly Valid '', i.e. the premise logically entails the conclusion: # Premise: 5 is both Even and Odd . (In our above formulation, this is P and not-P.) # Conclusion: God exists. (This is Q.) But atheists have no less reason to celebrate than theists, for ''this'' argument is ''also'' valid: # Premise: 5 is both even and odd. (This is P and not-P.) # 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. 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! Proof of the paradox 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: # (P and ¬P) is false. (See the Truth Table entry for ∧ .) # Therefore, (P and ¬P) → Q is Vacuously True . (See the Truth Table entry for → .) The second then, might interest those who find truth tables aesthetically flawed: # Suppose P and ¬P. Under this assumption we can derive: ## P ( Conjunction Elimination ) ## ¬P ( Conjunction Elimination ) ## Suppose ¬Q. Under this assumption we can derive: ### P (Copying from above) ## Thus ¬Q → P ( Conditional Proof ) ## ¬P → Q ( Contrapositive of previous line) ## Q ( Modus Ponens ) # Thus (P and ¬P) → Q ( Conditional Proof ) CONTRADICTIONS AND PHILOSOPHY 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.) Meta-contradiction 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. SEE ALSO |
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