上下偏量及其单调性--直接/间接量化与算子套叠

AT-LEAST/AT-MOST and Their Monotonicity:Direct/Indirect Quantization and Operator-overlapping

上/下偏量算子的性质是广义量词理论中研究的重要对象,但对它们的单调性计算尚无系统的讨论。当句子中只有一个算子时,广义量词理论较好地描写了其语义,也即上偏量算子左右单调上升,下偏量算子左右单调下降。但是,广义量词理论没有注意到下偏量与全称量化的单调下降有一个重要的差异:下偏量可能出现“空集反例”。另外,也没有注意到左单调是直接量化,右单调是间接量化,右单调的成分可以自由地加上新的算子。更为重要的是,当句中有两个算子套叠的时候,广义量词理论没有给出解决方案。实际上,外层成分的单调性由外层算子的左单调决定;内层成分则必须经历两步计算:先按照内层算子的左单调性质计算,再将计算结果输入外层算子的右单调,才能得到最终结果。

The properties of AT-LEAST/AT-MOST operators are important research objects in the theory of generalized quantifiers,but there is no systematic discussion on their monotonicity calculation.When there is only one operator in a sentence,the generalized quantifier theory describes its meaning well,that is,the AT-LEAST operator is left and right monotone increasing,while the AT-MOST operator is left and right monotone decreasing.However,it was not noticed that there is an important difference in the monotone decreasing between the AT-MOST operator and the universal quantization operator:the AT-MOST operator may exhibit“empty-set counterexamples”.Furthermore,it was not noticed that left-monotony is directly quantified and right-monotony is indirectly quantified.The right monotonic component can be freely added with new operators.More importantly,when two operators overlap in a sentence,the generalized quantifier theory does not provide a solution.In fact,the monotonicity of the outer components is determined by the left-monotonicity of the outer operator.The inner component must go through two steps of calculation:first,calculating according to the left monotonic property of the inner operator,and then inputting the calculation result into the right monotonic property of the outer operator to obtain the final result.