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The above example explained — in a simplified way — how a mathematician might weaken her conjecture in the face of counterexamples, but counterexamples can also be used to demonstrate the necessity of certain assumptions and hypothesis. For example, suppose that after a while, the mathematician above settled on the new conjecture "All ...
For example, in a study by Meyers-Levy and Maheswaran, subjects were more likely to counterfactual think alternative circumstances for a target person if their house burned down three days after they forgot to renew their insurance versus six months after they forgot to renew their insurance.
Objections to evolution have been raised since evolutionary ideas came to prominence in the 19th century. When Charles Darwin published his 1859 book On the Origin of Species, his theory of evolution (the idea that species arose through descent with modification from a single common ancestor in a process driven by natural selection) initially met opposition from scientists with different ...
This dispute may be better understood when considering specific examples, such as the "continuum hypothesis". The continuum hypothesis has been proven independent of the ZF axioms of set theory, so within that system, the proposition can neither be proven true nor proven false. A formalist would therefore say that the continuum hypothesis is ...
This can be done by a counter example of the same form of argument with premises that are true under a given interpretation, but a conclusion that is false under that interpretation. In informal logic this is called a counter argument. The form of an argument can be shown by the use of symbols.
Example 1. One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but an obviously false conclusion. For example: If someone lives in San Diego, then they live in California. Joe lives in California. Therefore, Joe lives in San Diego. There are many places to live in California other than San Diego.
Such a proof is again a refutation by contradiction. A typical example is the proof of the proposition "there is no smallest positive rational number": assume there is a smallest positive rational number q and derive a contradiction by observing that q / 2 is even smaller than q and still positive.
If the form of the contradiction is that we can derive a further counterexample D, that is smaller than C in the sense of the working hypothesis of minimality, then this technique is traditionally called proof by infinite descent. In which case, there may be multiple and more complex ways to structure the argument of the proof.