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The Epimenides paradox reveals a problem with self-reference in logic. It is named after the Cretan philosopher Epimenides of Knossos who is credited with the original statement. A typical description of the problem is given in the book Gödel, Escher, Bach, by Douglas Hofstadter:
Epimenides was a Cretan who made the immortal statement: "All Cretans are liars."
In logic, the law of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradiction, and the law of identity. However, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws.
In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied. In "this sentence is a lie" the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction.
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion. A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time. They result in "persistent contradiction between interdependent elements" leading to a lasting "unity of opposites".
The raven paradox, also known as Hempel's paradox, Hempel's ravens, or rarely the paradox of indoor ornithology, is a paradox arising from the question of what constitutes evidence for the truth of a statement. Observing objects that are neither black nor ravens may formally increase the likelihood that all ravens are black even though, intuitively, these observations are unrelated.
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.
Wigner's friend is a thought experiment in theoretical quantum physics, first published by the Hungarian-American physicist Eugene Wigner in 1961, and further developed by David Deutsch in 1985. The scenario involves an indirect observation of a quantum measurement: An observer observes another observer who performs a quantum measurement on a physical system. The two observers then formulate a statement about the physical system's state after the measurement according to the laws of quantum theory. In the Copenhagen interpretation, the resulting statements of the two observers contradict each other. This reflects a seeming incompatibility of two laws in the Copenhagen interpretation: the deterministic and continuous time evolution of the state of a closed system and the nondeterministic, discontinuous collapse of the state of a system upon measurement. Wigner's friend is therefore directly linked to the measurement problem in quantum mechanics with its famous Schrödinger's cat paradox.
Curry's paradox is a paradox in which an arbitrary claim F is proved from the mere existence of a sentence C that says of itself "If C, then F", requiring only a few apparently innocuous logical deduction rules. Since F is arbitrary, any logic having these rules allows one to prove everything. The paradox may be expressed in natural language and in various logics, including certain forms of set theory, lambda calculus, and combinatory logic.
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic". An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics.
Moore's paradox concerns the apparent absurdity involved in asserting a first-person present-tense sentence such as "It is raining, but I do not believe that it is raining" or "It is raining, but I believe that it is not raining." The first author to note this apparent absurdity was George E. Moore. These 'Moorean' sentences, as they have become known, are paradoxical in that while they appear absurd, they nevertheless
- Can be true;
- Are (logically) consistent; and
- Are not (obviously) contradictions.
The sorites paradox is a paradox that results from vague predicates. A typical formulation involves a heap of sand, from which grains are removed individually. With the assumption that removing a single grain does not cause a heap to become a non-heap, the paradox is to consider what happens when the process is repeated enough times that only one grain remains: is it still a heap? If not, when did it change from a heap to a non-heap?
In classical logic, intuitionistic logic and similar logical systems, the principle of explosion, or the principle of Pseudo-Scotus, is the law according to which any statement can be proven from a contradiction. That is, from a contradiction, any proposition can be inferred; this is known as deductive explosion.
Fitch's paradox of knowability is one of the fundamental puzzles of epistemic logic. It provides a challenge to the knowability thesis, which states that every truth is, in principle, knowable. The paradox is that this assumption implies the omniscience principle, which asserts that every truth is known. Essentially, Fitch's paradox asserts that the existence of an unknown truth is unknowable. So if all truths were knowable, it would follow that all truths are in fact known.
The barbershop paradox was proposed by Lewis Carroll in a three-page essay titled "A Logical Paradox", which appeared in the July 1894 issue of Mind. The name comes from the "ornamental" short story that Carroll uses in the article to illustrate the paradox. It existed previously in several alternative forms in his writing and correspondence, not always involving a barbershop. Carroll described it as illustrating "a very real difficulty in the Theory of Hypotheticals". From the viewpoint of modern logic, it is seen not so much as a paradox than as a simple logical error. It is of interest now mainly as an episode in the development of algebraic logical methods when these were not so widely understood, although the problem continues to be discussed in relation to theories of implication and modal logic.
The Boy or Girl paradox surrounds a set of questions in probability theory, which are also known as The Two Child Problem, Mr. Smith's Children and the Mrs. Smith Problem. The initial formulation of the question dates back to at least 1959, when Martin Gardner featured it in his October 1959 "Mathematical Games column" in Scientific American. He titled it The Two Children Problem, and phrased the paradox as follows:
A self-refuting idea or self-defeating idea is an idea or statement whose falsehood is a logical consequence of the act or situation of holding them to be true. Many ideas are called self-refuting by their detractors, and such accusations are therefore almost always controversial, with defenders stating that the idea is being misunderstood or that the argument is invalid. For these reasons, none of the ideas below are unambiguously or incontrovertibly self-refuting. These ideas are often used as axioms, which are definitions taken to be true, and cannot be used to test themselves, for doing so would lead to only two consequences: consistency or exception (self-contradiction).
The paradoxes of material implication are a group of true formulae involving material conditionals whose translations into natural language are intuitively false when the conditional is translated as "if ... then ...". A material conditional formula is true unless is true and is false. If natural language conditionals were understood in the same way, that would mean that the sentence "If the Nazis had won World War Two, everybody would be happy" is vacuously true. Given that such problematic consequences follow from a seemingly correct assumption about logic, they are called paradoxes. They demonstrate a mismatch between classical logic and robust intuitions about meaning and reasoning.
Stephen Yablo is a Canadian-born American philosopher. He is David W. Skinner Professor of Philosophy at the Massachusetts Institute of Technology (MIT) and taught previously at the University of Michigan, Ann Arbor. He specializes in the philosophy of logic, philosophy of mind, metaphysics, philosophy of language, and philosophy of mathematics.
The crocodile paradox, also known as crocodile sophism, is a paradox in logic in the same family of paradoxes as the liar paradox. The premise states that a crocodile, who has stolen a child, promises the parent that their child will be returned if and only if they correctly predict what the crocodile will do next.
References
- 1 2 3 4 5 6 7 Chow, T. Y. (1998). "The surprise examination or unexpected hanging paradox" (PDF). The American Mathematical Monthly. 105 (1): 41–51. arXiv: math/9903160 . doi:10.2307/2589525. JSTOR 2589525.
- ↑ Stanford Encyclopedia discussion of hanging paradox together with other epistemic paradoxes
- ↑ Binkley, Robert (1968). "The Surprise Examination in Modal Logic". The Journal of Philosophy. 65 (5): 127–136. doi:10.2307/2024556. JSTOR 2024556.
- ↑ Sorensen, R. A. (1988). Blindspots. Oxford: Clarendon Press. ISBN 978-0198249818.
- ↑ "Unexpected Hanging Paradox". Wolfram.
- ↑ Fitch, F. (1964). "A Goedelized formulation of the prediction paradox". Am. Phil. Q. 1 (2): 161–164. JSTOR 20009132.
- ↑ Chow, T. Y. (1998). "The surprise examination or unexpected hanging paradox" (PDF). The American Mathematical Monthly. 105 (1): 41–51. arXiv: math/9903160 . doi:10.2307/2589525. JSTOR 2589525. Archived from the original (PDF) on 7 December 2015. Retrieved 30 December 2007.
