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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.
An indirect question is expressed by changing the mood of the main verb from the indicative to the subjunctive. Some rhetoric questions change the verb to the accusative, followed by the infinitive, as if it were a real declarative statement in direct speech [17]). It is normally appropriate to retain the word that introduces the question, but ...
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 ...
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.
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 ".
Following Gottlob Frege and Bertrand Russell, Hilbert sought to define mathematics logically using the method of formal systems, i.e., finitistic proofs from an agreed-upon set of axioms. [5] One of the main goals of Hilbert's program was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.
In Latin, the sequence of tenses rule affects dependent verbs in the subjunctive mood, mainly in indirect questions, indirect commands, and purpose clauses. [4] If the main verb is in one of the non-past tenses, the subordinate verb is usually in the present or perfect subjunctive (primary sequence); if the main verb is in one of the past tenses, the subordinate verb is usually in the ...
In English, for example, the interrogative is supposed to indicate that the utterance is (intended as) a question; the directive indicates that the utterance is (intended as) a directive illocutionary act (an order, a request, etc.); the words "I promise" are supposed to indicate that the utterance is (intended as) a promise.
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