“analytic”的概念、定义、翻译、参考文献-科学参考

analytic

Mathematics

A function is analytic if it can be defined by a convergent Taylor series in a neighbourhood of any point of the domain. An analytic function necessarily has derivatives of all orders, but the converse is not true. The real function, defined by f(x) = exp(−1/x²) for x ≠ 0 and f(0) = 0, is infinitely differentiable at x = 0, with all derivatives being 0; thus, the Taylor series at x = 0 only converges to the function at 0. The space of analytic functions is denoted C^ω‎.

Logic

1. According to the theory of judgement of the philosopher Immanuel Kant (1724–1804), any judgement or inference in which the consequent is ‘contained in’ or ‘part of’ the content of the antecedent. The relationship of entailment in an analytic judgement is one in which inference proceeds from a literal conceptual analysis of the antecedent. For example, Kant counts the sentence:
- • All bachelors are unmarried.
as analytic because the concept corresponding to ‘bachelorhood’ is identical to the conjunction of the concepts of ‘maleness’ and ‘unmarried-ness’. Hence, to conclude that if $x$ is a bachelor then $x$ is unmarried is literally a ‘breaking apart’ of the concept of bachelorhood. In distinction to analytic judgements, a statement such as:
- • Most bachelors will become married.
is ampliative because its truth follows from facts not found within the concept of bachelorhood, e.g., social demographics, social trends, etc.
2. Describes any sentence that is true in virtue of the meanings of the words appearing in it, i.e., a sentence whose truth can be established by an analytic judgement in the foregoing sense.

3. In a propositional logic $L$ , describes conditional connectives $\to$ interpreted as capturing essential features of Kantian analytic judgements. Formally, analytic conditionals are those with the feature that a formula $φ \to ψ$ is an $L$ -theorem only if every atomic formula appearing as a subformula of $ψ$ is a subformula of $φ$ .
4. In proof theory, a proof in a sequent calculus that is cut free, that is, a proof in which no instance of the cut rule is applied.

单词	analytic
释义	analytic Mathematics A function is analytic if it can be defined by a convergent Taylor series in a neighbourhood of any point of the domain. An analytic function necessarily has derivatives of all orders, but the converse is not true. The real function, defined by f(x) = exp(−1/x²) for x ≠ 0 and f(0) = 0, is infinitely differentiable at x = 0, with all derivatives being 0; thus, the Taylor series at x = 0 only converges to the function at 0. The space of analytic functions is denoted C^ω‎. Logic 1. According to the theory of judgement of the philosopher Immanuel Kant (1724–1804), any judgement or inference in which the consequent is ‘contained in’ or ‘part of’ the content of the antecedent. The relationship of entailment in an analytic judgement is one in which inference proceeds from a literal conceptual analysis of the antecedent. For example, Kant counts the sentence: • All bachelors are unmarried. as analytic because the concept corresponding to ‘bachelorhood’ is identical to the conjunction of the concepts of ‘maleness’ and ‘unmarried-ness’. Hence, to conclude that if $x$ is a bachelor then $x$ is unmarried is literally a ‘breaking apart’ of the concept of bachelorhood. In distinction to analytic judgements, a statement such as: • Most bachelors will become married. is ampliative because its truth follows from facts not found within the concept of bachelorhood, e.g., social demographics, social trends, etc. 2. Describes any sentence that is true in virtue of the meanings of the words appearing in it, i.e., a sentence whose truth can be established by an analytic judgement in the foregoing sense. 3. In a propositional logic $L$ , describes conditional connectives $\to$ interpreted as capturing essential features of Kantian analytic judgements. Formally, analytic conditionals are those with the feature that a formula $φ \to ψ$ is an $L$ -theorem only if every atomic formula appearing as a subformula of $ψ$ is a subformula of $φ$ . 4. In proof theory, a proof in a sequent calculus that is cut free, that is, a proof in which no instance of the cut rule is applied.
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