Self-refuting ideas are ideas or statements whose falsehood is a logical consequence of the act or situation of holding them to be true. Many ideas are accused by their detractors of being self-refuting, and such accusations are therefore almost always controversial, with defenders claiming that the idea is being misunderstood or that the argument is invalid. For these reasons, none of the ideas below is unambiguously or incontrovertibly self-refuting.
Directly self-denying statements
The Epimenides paradox is an instance of a statement of the form "this statement is false". Such statements troubled philosophers, especially when there was a serious attempt to formalize the foundations of logic. Bertrand Russell developed his "Theory of Types" to formalize a set of rules which would prevent such statements (more formally Russell's paradox) being made in symbolic logic. This work has led to the modern formulation of axiomatic set theory. While Russell's formalization didn't contain such paradoxes, Kurt Gödel showed that it must contain independent statements. Any logical system that is rich enough to contain elementary arithmetic contains propositions whose interpretation is "this proposition is unprovable" (from within the logical system concerned), and hence no such system can be both complete and consistent.
Indirectly self-denying statements or "fallacy of the stolen concept"
Objectivists define the fallacy of the stolen concept which consists of the act of using a concept while ignoring, contradicting or denying the validity of the concepts on which it logically and genetically depends. A claimed example of the stolen concept fallacy is anarchist Pierre-Joseph Proudhon's assertion, "All property is theft".
- While discussing the hierarchical nature of knowledge, Nathaniel Branden states, “Theft” is a concept that logically and genetically depends on the antecedent concept of “rightfully owned property”—and refers to the act of taking that property without the owner’s consent. If no property is rightfully owned, that is, if nothing is property, there can be no such concept as “theft.” Thus, the statement “All property is theft” has an internal contradiction: to use the concept “theft” while denying the validity of the concept of “property,” is to use “theft” as a concept to which one has no logical right—that is, as a stolen concept.
Others have said the slogan is not an instance of the stolen concept fallacy under Proudhon's intended meaning. Proudhon used the term "property" with reference to claimed ownership in land, factories, etc. He believed such claims were illegitimate, and thus a form of theft from the commons.  Proudhon explicitly states that the phrase "property is theft" is analogous to the phrase "slavery is murder". According to Proudhon, the slave, though biologically alive, is clearly in a sense "murdered". The "theft" in in his terminology does not refer to ownership anymore than the "murder" refers directly to physiological death, but rather both are meant as terms to represent a denial of specific rights. 
- To count as rational, a belief must be freely chosen, which according to the determinist is impossible
- Any kind of debate seems to be posited on the idea that the parties involved are trying to change each others minds.
Both arguments can be countered, for instance:
- A belief is freely chosen if it is chosen without duress (according to compatibilism).
- Determinism doesn't assert that people never change their minds, only that such changes are necessitated by causes. If someone changes their mind as the result of hearing an argument, that was a cause.
It has been argued that extreme ethical egoism is self-defeating. Faced with a situation of limited resources, egoists would consume as much of the resource as they could, making the overall situation worse for everybody. Egoists rejoin that if the situation becomes worse for everybody, that would include the egoist, so it is not in fact in their rational self-interest to take things to such extremes.
The philosopher Mary Midgley claims the idea that "nothing exists except matter" is also self-refuting because if it were true neither it, nor any other idea, would exist, and similarly that an argument to that effect would be self-refuting because it would deny its own existence.
However, other forms of materialism may escape this kind of argument because, rather than eliminating the mental, they seek to identify it with, or reduce it to, the material.. For instance, identity theorists such as J. J. C. Smart, Ullin Place and E. G. Boring claim that ideas exist materially as patterns of neural structure and activity.
The first notable self-refuting idea is the Epimenides paradox, a statement attributed to Epimenides, a Cretan philosopher, that "All Cretans are Liars". Interpreted (for the present purpose) as meaning "no Cretan ever speaks the truth" this cannot be true if uttered by a Cretan.
This is a particularly contentious proposal: Alvin Plantinga argues in his Evolutionary argument against naturalism that the combination of Naturalism and Evolution is "in a certain interesting way self-defeating" because if it were true there would be insufficient grounds to believe that human cognitive faculties are reliable. This argument has been supported and criticised by a variety of thinkers
- Everything is caused by something other than itself
- Therefore the universe was caused by something other than itself.
- The string of causes cannot be infinitely long.
- If the string of causes cannot be infinitely long, there must be a first cause.
- Therefore, there must be a first cause, namely god.
– and suggests that this is self-refuting because "if everything has a cause other than itself, then god must have a cause other than himself. But if god has a cause other than himself, he cannot be the first cause. So if the first premise is true, the conclusion must be false."
The Philosopher Anthony Kenny argues that the idea, "common to theists like Aquinas and Descartes and to an atheist like Russell" that "Rational belief [is] either self-evident or based directly or indirectly on what is evident" (which he termed "foundationalism" following Plantinga) is self-refuting on the basis that this idea is itself neither self-evident nor based directly or indirectly on what is evident and that the same applies to other formulations of such foundationalism. However, the self-evident impossibility of infinite regress can be offered as a justification for foundationalism. Following the identification of problems with "naive foundationalism", the term is now often used re-defined to focus on incorrigible beliefs (modern foundationalism), or basic beliefs (reformed foundationalism).
Biologist PZ Myers, commenting on Richard Dawkins' book The God Delusion, has suggested that "postulating an immensely complicated being to explain the creation of an immensely complicated universe doesn't actually explain anything and is self-refuting." Theistic philosophers, however, contend that this mis-states the First Cause argument and that such characterizations depart from the standard understanding of God as epitomized in the work of philosophers such as Augustine, Anselm, and Aquinas: God's classical attribute of aseity instead postulates a simple being. However, it is not clear how such a priori argumentation is to be combined with natural theology.
It is often asserted that relativism about truth must be applied to itself.. The cruder form of the argument concludes that since the relativist is asserting relativism as an absolute truth, it leads to a contradiction. Relativists often rejoin that in fact relativism is only relatively true, leading to a subtler problem: the absolutist, the relativist's opponent, is perfectly entitled, by the relativist's own standards, to reject relativism. That is, the relativist's arguments can have no normative force over someone who has different basic beliefs.
Skeptics claim "nothing can be known". Can that claim itself be known, or is it self-refuting?One very old response to this problem is Academic skepticism: an exception is made for the skeptics own claim. This leads to further debate about consistency and special pleading. Another response is to accept that "nothing can be known" cannot itself be known, so that it is not known whether anything is knowable or not. This is Pyrrhonic skepticism.
The statement "no statements are true unless they can be proven scientifically", is claimed to be self-refuting insofar as it cannot be proven scientifically; the same goes for essentially similar views like "no statements are true unless they can be shown empirically to be true". (This kind of issue was a serious problem for logical positivism).
On the face of it, a statement of solipsism is self-defeating, because a statement assumes another person to whom the statement is made. (That is to say, an unexpressed private belief in solipsism is not self-refuting). The solipsist can adopt the rather surreal maneuver of claiming that their interlocutor is in fact a figment of their imagination, but since their interlocutor knows they are not, they are not going to be convinced!
Anthony Kenny also argues that utilitarianism is self-refuting on the grounds that either determinism is true or false. If it is true, then we have no choice over our actions. But if it is false then the consequences of our actions are unpredictable, not least because they will depend on the actions of others whom we cannot predict This would be refuted by the truth of compatibilism.
Verification- and falsification-principles
The statements "statements are meaningless unless they can be empirically verified" and "statements are meaningless unless they can be empirically falsified" are both claimed to be self-refuting on the basis that they can neither be empirically verified nor falsified
"My propositions are elucidatory in this way: he who understands me finally recognizes them as senseless, when he has climbed out through them, on them, over them. (He must so to speak throw away the ladder, after he has climbed up on it".(6.54)
Notes and References
- Russell B, Whitehead A.N., Principia Mathematica
- The Stolen Concept by Nathaniel Branden - originally published in The Objectivist Newsletter in January 1963.
- Rockwell, L. Performative Contradicitons and Subtle Misunderstandings
- "Second, the argument for [determinisim is self defeating. A determinist must contend that both he and the nondeterminist are determined to believe what they believe. Yet the determinist attempts to convince the nondeterminist that determinism is true and thus ought to be believed. However, on the basis of pure determinism "ought" has no meaning. For "ought" means "could have and should have done otherwise." But this is impossible according to determinism. A way around this objection is for the determinist to argue that he was determined to say that one ought to accept his view. However, his opponent can respond by saying that he was determined to accept a contrary view. Thus determinism cannot eliminate an opposing position. This allows the possibility for a free will position." Believe
- "Determinism is self-defeating. A determinist insists that both determinists and non-determinists are determined to believe what they believe. However, determinists believe self-determinists are wrong and ought to change their view. But "ought to change" implies they are free to change, which is contrary to determinism." 
- http://www.britannica.com/eb/article-252577/ethics Brittanica
- see Mary Midgley The Myths we Live by
- Baker, L. (1987). Saving Belief Princeton, Princeton University Press
- Reppert, V. (1992). Eliminative Materialism, Cognitive Suicide, and Begging the Question. Metaphilosophy 23: 378-92.
- Boghossian, P. (1990). The Status of Content Philosophical Review 99: 157-84. and (1991)The Status of Content Revisited. Pacific Philosophical Quarterly 71: 264-78.
- Hill, C. Identity Theory
- "To the author a perfect correlation is identity. Two events that always occur together at the same time in the same place, without any temporal or spatial differentiation at all, are not two events but the same event. The mind-body correlations as formulated at present, do not admit of spatial correlation, so they reduce to matters of simple correlation in time. The need for identification is no less urgent in this case." Place, U.T., Identity Theories in A Field Guide to the Philosophy of Mind. Società italiana per la filosofia analitica. Marco Nanni (ed.). ((online))
-  Dictionary of the Philosophy of Mind
- Alvin Plantinga in Naturalism Defeated? Ed James Beilby Cornell University Press 2002 p p
- John Polkinghorne is an example of a scientist-theologian who is supportive of Plantinga's position
- Fitelson, B. and Sober, E.Plantinga’s Probability Arguments Against Evolutionary Naturalism
- Robbins, J. Evolutionary Naturalism, Theism, and Skepticism about the External World
- see eg Naturalism Defeated? Ed James Beilby Cornell University Press 2002
- Schick, Theodore. "The 'Big Bang' Argument for the Existence of God". Retrieved 2007-05-29.
- Anthony Kenny What is Faith? Oxford: OUP 1992 ISBN 0192830678 pp9-10. This particular chapter is based on a 1982 lecture which may explain the shift in the meaning of the term "foundationalism" since then
- Stanford Encyclopedia of Philosophy on foundationalism
- Myers, Paul Zachary. "Open Letter to H. Allen Orr". Retrieved 2007-04-16.
- Brower, Jeffrey E. "Simplicity and Aseity (Forthcoming in Oxford Handbook of Philosophical Theology)" (PDF). Retrieved 2007-05-01.
- Cognitive Relativism, Internet Encyclopedia of Philosophy
- The problem of self-refutation is quite general. It arises whether truth is relativized to a framework of concepts, of beliefs, of standards, of practices.[http://www.science.uva.nl/~seop/entries/relativism/ Stanford Encyclopedia of Philosophy
- "If truth is relative, then non-relativist points of view can legitimately claim to be true relative to some standpoints." Westacott, E. On the Motivations for Relativism
- The Gallilean Library
- Suber, P. Classical Skepticism
- Internet Encyclopedia of Philosophy
- see eg Keith Ward, Is Religion Dangerous?
- ""As against solipsism it is to be said, in the first place, that it is psychologically impossible to believe, and is rejected in fact even by those who mean to accept it. I once received a letter from an eminent logician, Mrs. Christine Ladd Franklin, saying that she was a solipsist, and was surprised that there were no others. Coming from a logician and a solipsist, her surprise surprised me." (Russsel, B., Human Knowledge: Its Scope and Limitsp. 180)."
- Anthony Kenny What I Believe
- see eg. the discussion by William P Alston in The Rationality of Theism (ISBN 0415263328) pp 26-34