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1、外文文献译文Syntax and semanticsA formal language usually requires a set of formation rulesi.e., a complete specification of the kinds of expressions that shall count as well-formed formulas (sentences or meaningful expressions), applicable mechanically, in the sense that a machine could check whether a c
2、andidate satisfies the requirements. This specification usually contains three parts: (1) a list of primitive symbols (basic units) given mechanically, (2) certain combinations of these symbols, singled out mechanically as forming the simple (atomic) sentences, and (3) a set of inductive clauses ind
3、uctive inasmuch as they stipulate that natural combinations of given sentences formed by such logical connectives as the disjunction “or,” which is symbolized “”; “not,” symbolized “”; and “for all ,” symbolized “(),” are again sentences. “()” is called a quantifier, as is also “there is some ,” sym
4、bolized “()”. Since these specifications are concerned only with symbols and their combinations and not with meanings, they involve only the syntax of the language.An interpretation of a formal language is determined by formulating an interpretation of the atomic sentences of the language with regar
5、d to a domain of objects., by stipulating which objects of the domain are denoted by which constants of the language and which relations and functions are denoted by which predicate letters and function symbols. The truth-value (whether “true” or “false”) of every sentence is thus determined accordi
6、ng to the standard interpretation of logical connectives. For example, p q is true if and only if p and q are true. (Here, the dot means the conjunction “and,” not the multiplication operation “times.”) Thus, given any interpretation of a formal language, a formal concept of truth is obtained. Truth
7、, meaning, and denotation are semantic concepts.If, in addition, a formal system in a formal language is introduced, certain syntactic concepts arise namely, axioms, rules of inference, and theorems. Certain sentences are singled out as axioms. These are (the basic) theorems. Each rule of inference
8、is an inductive clause, stating that, if certain sentences are the orems, then another sentence related to them in a suitable way is also atheorem. If p and “either not-p or q” (p q) are theorems, for example, then q is a theorem. In general, a theorem is either an axiom or the conclusion of a rule
9、of inference whose premises are theorems.In 1931 Gdel made the fundamental discovery that, in most of the interesting (or significant) formal systems, not all true sentences are theorems. It follows from this finding that semantics cannot be reduced to syntax; thus syntax, which is closely related t
10、o proof theory, must often be distinguished from semantics, which is closely related to model theory. Roughly speaking, syntax，as conceived in the philosophy of mathematics，is a branch of number theory, and semantics is a branch of set theory, which deals with the nature and relations of aggregates.
11、Historically, as logic and axiomatic systems became more and more exact, there emerged, in response to a desire for greater lucidity, a tendency to pay greater attention to the syntactic features of the languages employed rather than to concentrate exclusively on intuitive meanings. In this way, log
12、ic, the axiomatic method (such as that employed in geometry), and semiotic (the general science of signs) converged toward metalogic.Truth definition of the given languageThe formal system N admits of different interpretations, according to findings of Gdel (from 1931) and of the Norwegian mathemati
13、cian Thoralf Skolem, a pioneer in metalogic (from 1933). The originally intended, or standard, interpretation takes the ordinary nonnegativeintegers 0, 1, 2, . . . as the domain, the symbols 0 and 1 as denoting zero and one, and the symbols + and as standing for ordinary addition and multiplication.
14、 Relative to this interpretation, itis possible to give a truth definition of the language of N.It is necessary first to distinguish between open and closed sentences. An open sentence, such as x = 1, is one that may be either true or false depending on the value of x, but a closed sentence, such as
15、 0 = 1 and (x) (x = 0) or “All xs are zero,” is one that has a definite truth-valuein this case, false (in the intended interpretation).1. A closed atomic sentence is true if and only if it is true in the intuitive sense; for example, 0 = 0 istrue, 0 + 1 = 0 is false.This specification as it stands
16、is not syntactic, but, with some care, it is possible to give an explicit and mechanical specification of those closed atomic sentences that are true in the intuitive sense.2. A closed sentence A is true if and only if A is not true.3. A closed sentence A B is true if and only if either A or B is tr
17、ue.4. A closed sentence ()A() is true if and only if A() is true for every value of i.e., if A(0), A(1), A(1 + 1), . . . are all true.The above definition of truth is not an explicit definition; it is an inductive one. Using concepts fromset theory, however, it is possible to obtain an explicit defi
18、nition that yields a set of sentences that consists of all the true ones and only them. If Gdels method of representing symbols and sentences by numbers is employed, it is then possible to obtain in set theory a set of natural numbers that are just the Gdel numbers of the true sentences of N.There i
19、s a definite sense in which it is impossible to define the concept of truth within a language itself. This is proved by the liar paradox: if the sentence “I am lying,” or alternatively(1) This sentence is not true.is considered, it is clearsince (1) is “This sentence”that if (1) is true, then (1) is
20、 false; on the other hand, if (1) is false, then (1) is true. In the case of the system N, if the concept of truth were definable in the system itself, then (using a device invented by Gdel) it would be possible to obtain in“ N ”a sentence that amounts to (1) and that thereby yields a contradiction.
21、外文文献原文语法和语义一份正式的语言通常需要一套形成,齐全的规格的种类中,作为规范的表达方式,要计算公式(句子或有意义的表达),适用的机械,在这个意义上说,机器就会检查是否满足要求的候选人。本规格书通常包含三个部分:(1)一个列表的原始的符号(基本单元)给机械,(2)特定的组合,这些符号,特别强调了机械成形(原子)的简单句子,以及(3)一组感应条款-感应鉴于二者规定自然的组合形成的语态句这样的逻辑篇章脱节.因为这些规格是只关心与符号及其组合,而不是与意义,他们仅仅包括语法的语言。语言的一个解释是由一个正式制定的解释语言的原子句关于某个领域的objects.,通过规定哪些物体的领域得到用这常数之
22、语言与这关系和功能是通过引入,谓词的字母和功能的符号。(是否的应着重于“对”或“否”)的每一个句子是按照这样的标准解释逻辑篇章。例如,p问是真的,当且仅当p和q是真实的。(在这里,点手段的结合”,“不是一个令人讨厌的乘法运算”的时代。)因此,给出任何解释一套正式的语言,一种是真理的形式概念设计提供了依据。真理、意义以及外延是语义的概念。此外,如果一个正式的制度形式语言的概念,介绍了某种特定的句法规则,公理,即产生的推论,和定理。某些句子都是选作公理系统。这些都是(基本)定理。每一个规则的推理是一种感应条款,说明,如果某些句子相似定理,然后另一个句子与他们相关的是在一种合适的方式也atheore
23、m。如果p和q”或“要么not-pp(q)是定理,举个例子,然后问是一个定理。一般来说,一个定理要么是一个公理或结论法治的推理的前提定理。1931年哥德尔做出了基本的发现,在大部分的有趣(或重要)正规制度,并不是所有真正的句子都是定理。有发现,不可能被简化到句法语义语法,这是,因此密切相关理论,常常需要证明,这是区别于语义模型理论密切相关。大致来说,语法,作为孕育于哲学的一个分支,是数学数论、和语义集合论的一个分支,这涉及到自然及两者的关系蕴。从历史上看,逻辑和公理系统变得越来越精确,出现了,这是为了回应明朗了,想要更多地倾向于更加关注人的句法特点的语言雇佣而不是专注于直观的意义。通过这种方式
24、,逻辑,在公理化方法(如,雇用了几何),和符号(一般科学向元逻辑)聚合的迹象。待添加的隐藏文字内容2给定语言的真理定义正式制度N承认的不同的解释,根据调查结果,从1931年拍摄)和哥德尔(挪威数学家Thoralf先驱、斯柯林元逻辑(从1933年)。原本打算的,或者标准的,普通的解释体制 0、1、2、。是本领域,0和1作为符号表示零的一个,和标志+和成站立为普通的加法和乘法。相对于这个解释的话,那么可以比较真实的语言定义的N。有必要先区分开启和关闭的句子。一个开放的句子,如x = 1,是一种,可能为真或假根据x的价值,而是一种封闭的句子,如0 = 1和(x)(x = 0),或“所有的x是零”,是
25、假(预期解释)，在这种情况下指一个明确的真值。1。一个封闭的原子的句子是真的当且仅当它是真的在直觉上的观念;例如,0 = 0为真,0 + 1 = 0是假的。本规格书正如眼下的不是句法,但是,跟一些照顾,就有可能给出一个明确的和机械规格的原子的句子是真的关闭在直觉上说得过去。2。一个封闭的句子一句是真的当且仅当一个是不正确的3。一个封闭的判B是真的,当且仅当要么A或B是真实的。4.一个封闭的句子()A() 当且仅当对于每一个i的值都是真的：A(0), A(1), A(1 + 1), . . .全真上述定义的真相并不是一个显式定义;它是一种感应。利用概念fromset理论,然而,它有可能获得一个显式定义能够产生一个的句子,由所有的真实的故事,只有他们。如果哥德尔的符号和句子的表达方式由编号被录用,然后你就可以获得在集合论一套自然的数字,只是哥德尔的编号的真正的句子N。有明确的意义上,它是不可能在界定概念真理的一种语言本身。这是证明的说谎者悖论:如果这句话我说谎,”或干脆：“(1)这句话是不正确的。”被认为是,很明显的,因为(1),是“这句话里的 -如果(1)是真实的,那么(1)是虚假的;另一方面,如果(1)是假的,然后是(1)是真实的。在这个例子中,如果系统N的概念等等真理,都是系统本身,然后(利用哥德尔发明的装置) 在“N”来获得一个句子这样多的(1),从而产生一个矛盾是可能的。
