目前关于预设的真值语义学研究主要是各种三值语义。这看起来有自然的一面，但其实有一个严重的误区，将“无定义”也看成一个与预设原句真值处于同一层面的真值，混淆了对象语言与元语言的区分。预设句与原句是不同层次的句子。本文试以专名指称存在预设为例，通过部分模型以及在此之上的同底扩张模型方法构造了复合模型，给出了一个既保证不同语言层次的区分又可以跨层次统一谈论原句和预设句关系的形式语义。通过这个语义，揭示了预设句与原句的2//[2]+[0]式的真值结构，使得通常情况下关于预设的默认得到形式上的刻画和呈现。%The research tradition on presupposition under the truth-value semantics requires that the presupposition to be true should be a necessary condition for the original sentence to be true. The trivalent semantics is one of the most typical methods under this tradition. It is natural at one point. However, there is a serious misunderstanding which takes the“undefinedness”as a truth-value at the same level as the truth-value of the original sentence, which ignores the difference between the object language and meta language. The original sentence and its presupposition belong to different language layers. Taking the referential presupposition triggered by proper names as an example, this gives a new formal semantic frame, using which could build a compound model that not only guarantees to differentiate the original sentence and its presupposition belong to different language layers, but also unifiedly talks about them across the layers. We have built formal languages L+and L, and their corresponding models M+and M (a partial model where some constant or variable may have no value), which are used to character-ize the truth semantic relationship between the original sentence and the presupposition. The former is an expansion of the latter so that they have the same base, which will make sure that all the constants and variables have appeared in M will also appear in M+. And M+ will also introduce a special kind of predicate E to describe an individual is exist in the meta language, which cannot be interpreted in M, and also makes sure that all these kind of individuals also belong to the domain of M. The domain of M is a subset of M+. The new semantic frame revels the truth-value structure of the original sentence and the presupposition as 2//[2]+[0], which makes the general understanding of presupposition to be formally described and presented.
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