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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.
An indirectly self-referential sentence would replace the phrase "this sentence" with an expression that effectively still referred to the sentence, but did not use the pronoun "this." An example will help to explain this. Suppose we define the quine of a phrase to be the quotation of the phrase followed by the phrase itself. So, the quine of:
However, indirect methods such as proof by contradiction can also be used with contraposition, as, for example, in the proof of the irrationality of the square root of 2. By the definition of a rational number , the statement can be made that " If 2 {\displaystyle {\sqrt {2}}} is rational, then it can be expressed as an irreducible fraction ".
Since the expression on the left is an integer multiple of 2, the right expression is by definition divisible by 2. That is, a 2 is even, which implies that a must also be even, as seen in the proposition above (in #Proof by contraposition). So we can write a = 2c, where c is also an integer.
A formal expression is a kind of string of symbols, created by the same production rules as standard expressions, however, they are used without regard to the meaning of the expression. In this way, two formal expressions are considered equal only if they are syntactically equal, that is, if they are the exact same expression.
For example, one might conjecture that a differential operator ought to satisfy a certain boundedness condition "for nice test functions," or one might state that some interesting topological invariant should be computable "for nice spaces X." object Anything that can be assigned to a variable and for which equality with another object can be ...
Reductio ad absurdum, painting by John Pettie exhibited at the Royal Academy in 1884. In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity") or apagogical arguments, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.
An indirect statement or question can replace the direct object of a verb that is related to thought or communication. An indirect statement is expressed by changing the case of the subject noun phrase from nominative to accusative and by replacing the main verb with an infinitive (as in the English phrase "You believe me to be a traitor" above).
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