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In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. [1] For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are ...
Frankfurt's examples are significant because they suggest an alternative way to defend the compatibility of moral responsibility and determinism, in particular by rejecting the first premise of the argument. According to this view, responsibility is compatible with determinism because responsibility does not require the freedom to do otherwise.
Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.. In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) have defined a wide variety of topological properties.
The assumption that if there is a counterexample, there is a minimal counterexample, is based on a well-ordering of some kind. The usual ordering on the natural numbers is clearly possible, by the most usual formulation of mathematical induction; but the scope of the method can include well-ordered induction of any kind.
The Gettier problem, in the field of epistemology, is a landmark philosophical problem concerning the understanding of descriptive knowledge.Attributed to American philosopher Edmund Gettier, Gettier-type counterexamples (called "Gettier-cases") challenge the long-held justified true belief (JTB) account of knowledge.
offering a modified assertion that definitionally excludes a targeted unwanted counterexample; using rhetoric to signal the modification; An appeal to purity is commonly associated with protecting a preferred group. Scottish national pride may be at stake if someone regularly considered to be Scottish commits a heinous crime.
Substituting v 1 into the identity and removing common factors gives the numerical example cited above. In 1988, Roger Frye found the smallest possible counterexample + + = for k = 4 by a direct computer search using techniques suggested by Elkies. This solution is the only one with values of the variables below 1,000,000.
One famous counterexample in topology is the Alexander horned sphere, showing that topologically embedding the sphere S 2 in R 3 may fail to separate the space cleanly. As a counterexample, it motivated mathematicians to define the tameness property, which suppresses the kind of wild behavior exhibited by the horned sphere, wild knot , and ...