AT-LEAST/AT-MOST and Their Monotonicity:Direct/Indirect Quantization and Operator-overlapping
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.
AT-LEASTAT-MOSTmonotonicitysentence with single operatoroperator-overlapping
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