On Axiomatic Systems for Arbitrary Systems of Sentences
On Axiomatic Systems for Arbitrary Systems of Sentences
复制标题
论任意句子系统的公理系统
DOI：
10.1007/978-3-0346-0145-0_2
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发表时间：
2012
影响因子：
1.4
通讯作者：
J. Legris
中科院分区：
文献类型：
--
作者：
P. Hertz;J. Legris
只要一个句子系统被认为是有效的，通常就没有必要把每一句话都背到记忆中去；只需选择其中的一些句子，然后其他句子就可以遵循了。众所周知，这样的句子被称为公理。这些公理的选择在一定程度上是武断的。然而，人们可能会问，一个句子系统具有几个公理系统的性质是否与其他显著的性质相互关联，以及是否有系统的方法来寻找包含最少可能句子的公理系统。在下文中，应传达一些想法，这些想法可能对处理这些问题或相关问题的前期工作有用。事实上，实际的利益问题是如此错综复杂，以至于最初似乎满足于极大的简化：我们只考虑某种类型的句子，我们可以象征性地写下的句子：(A1，。。。，an)→b，它可以通过如下公式在语言上表示：IF(A1，.。。此外，在本第一部分中将引入第二个简化，只考虑a→b类型的句子；然而，我们将在下一部分中将自己从这一限制中解放出来。此外，我们假设从某些句子中其他句子遵循的规则：因此，例如，句子a→b，b→c的有效性应该导致句子a→c的保持。然而，这种句子的实际意思、符号→在字符组合中的意思a→b或相应语言公式中的单词‘if’不必在此指示。在这一点上，我们的任务不是列出我们从哪里获得推理规则，这些句子在什么意义上出现在日常生活中，以及我们的问题可能与科学学科的结构有什么关系。此外，这里不能给出原因，为什么我们的考虑甚至没有开始达到这样的概括性水平，如果我们想通过它们来充分理解数学或物理句子和
Whenever a system of sentences is recognized to be valid, it is often not necessary to convey each and every sentence to memory; it is sufficient to choose some of them from which the rest can follow. Such sentences, as it is generally known, are called axioms. The choice of these axioms is to a certain degree arbitrary. One can ask, however, if the property of a system of sentences to have several axiom systems is interconnected with other remarkable properties, and if there are systematic approaches to find, as the case may be, that axiomatic system which contains the least possible number of sentences. In the following some thoughts shall be communicated, which might be useful as a pre-stage for the treatment of these or related problems. In fact the actual problem of interest is so entangled, that initially it seems appropriate to be content with an immense simplification: We only consider sentences of a certain type, sentences that we can write symbolically: (a1, . . . , an) → b and that can be expressed linguistically by formulations such as: If (a1, . . . , an) altogether holds, so does b. In addition, a second simplification will be introduced in the present first part, by only considering sentences of type a → b; however, we will liberate ourselves from this limitation in a following part. Further we assume rules according to which from certain sentences other ones follow: So, e.g., the validity of the sentences a → b, b → c should result in the holding of the sentence a → c. However, what is actually meant by such a sentence, what the symbol → means in the combination of characters a → b or the word ‘if’ in the corresponding linguistic formulation, does not have to be indicated here. At this point, it cannot be our task to lay out where we take the inference rules from, in what sense such sentences appear in ordinary life and which relation our question might have with the configuration of a scientific discipline. Also, the reason cannot be given here, why our considerations do not even begin to reach that level of generality which would be necessary if we wanted to reach by them a full understanding of the connection between mathematical or physical sentences and