Fuzzy Sets are Sets— A Reply to Goertz and Mahoney
Fuzzy Sets are Sets— A Reply to Goertz and Mahoney
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模糊集就是集合——对戈尔茨和马奥尼的答复
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发表时间：
2013
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通讯作者：
Q. Carsten
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作者：
Q. Carsten
首先，也许也是最重要的是，根据其定义，所有集合都在成员和非成员案件之间建立了质的区别。这也适用于模糊集合；否则，它们就不是集合。模糊集增加这种定性区别的唯一信息是，案例可以根据它们是(不是)集合成员的程度进行排序。事实上，我们认为G&M的“定性研究=集合论研究”命题的合理性很大程度上源于这样一个事实，即集合反映了案例的定性属性，而定性研究人员对案例的定性属性及其关系感兴趣。想一想“定性的”和“定量的”这两个词的拉丁语来源。量子的意思是“多少”或“多少”。对物品的评估是关于它们拥有多少特定财产。相反，QUIIS一词被翻译为“什么”或“属于哪种/种类/性质”。关于对象的定性陈述会在给定集合中的对象和外部对象之间建立质的差异。我们认为集合的这一性质既没有争议，也是定性研究的基础，因为它植根于集合论。有了清爽的布景，所有这些似乎都没有争议。案件既有正式成员资格，也有完全非成员资格，排除中间规则方便地排除了这两种相互排斥和共同详尽的定性状态之间的任何混淆，在这种情况下，案件可能是。看起来令人困惑的是，模糊集的性质--部分隶属度分数和排除中位数规则的崩溃--是否不可避免地意味着：(A)不能在案件之间进行定性区分；以及(B)应允许同一案件在两个或更多个相互排斥的集合中保持高成员资格。我们认为(A)模糊集首先表示哪些案例在性质上相同和不同，只有在此之后案例才是给定集合的成员；以及(B)案例不能同时是两个或更多集合的良好经验实例，正是因为研究人员认为存在两个或更多质上不同的现象值得给予不同的名称。应用于G&M的例子，如果我们认为民主不仅在性质上不同于独裁，而且不同于无政府主义，因此我们在文献中引入了这个新术语，那么对于我们来说，完全相同的案例的经验属性(例如，政体得分为7)将一个案例限定为两个概念上不同的集合(例如，民主和无政府主义)的良好实例，对我们来说似乎是令人困惑的。我们不确定G&M在这两个问题上的确切定位。他们确实认为，两个或更多个互斥集合中的部分成员资格应该是可能的，并且他们没有问题，在一个以上的集合中，这种部分成员资格高于0.5。我们同意，互斥集合中的部分成员资格是没有问题的--然而，当且仅当案例在多于一个集合中的成员资格不超过0.5。这就是我们不同意G&M的地方。这允许两种解释。G&M可能会声称，模糊集不会建立任何定性的区别，一切都只是一个程度问题。由于上述原因，这项研究将在2013年春季进行定性和多方法研究
First, and perhaps most importantly, all sets, by their very definition, establish a qualitative distinction between those cases that are members and those that are not. This also applies to fuzzy sets; otherwise, they would not be sets. The only information fuzzy sets add to this qualitative distinction is that cases can be ranked as to how much they are (not) members of a set. As a matter of fact, we think that much of the plausibility of G&M’s postulate “qualitative research = settheoretic research” precisely stems from the fact that sets reflect qualitative properties of cases and qualitative researchers are interested in, well, the cases’ qualitative properties and their relations. Consider the Latin origin of the words “qualitative” and “quantitative,” respectively. Quantum is translated as “how many” or “how much.” Objects are assessed as to how much of a certain property they possess. The word qualis, instead, is translated as “what” or “of what kind/sort/nature.” Qualitative statements about objects establish qualitative differences between objects that are in a given set and and those that are out. We deem this property of sets both uncontroversial and constitutive for the vision of qualitative research as being rooted in set theory. With crisp sets, all this seems pretty uncontroversial. Cases have either full membership or full non-membership, and the Rule of the Excluded Middle conveniently rules out any confusion between these two mutually exclusive and jointly exhaustive qualitative states in which cases can be. What seems confusing is whether or not the properties of fuzzy sets— partial membership scores and the breakdown of the Rule of the Excluded Middle—unavoidably mean that (a) qualitative distinctions between cases cannot be made; and, relatedly, that (b) one and the same case should be allowed to hold high membership in two or more mutually exclusive sets. We believe that (a) fuzzy sets first and foremost express which cases are qualitatively identical and different, and only afterwards to which degree cases are members of a given set; and (b) that cases cannot simultaneously be good empirical instances of two or more sets that have been created precisely because researchers believe that there are two or more qualitatively distinct phenomena worth being given different names. Applied to G&M’s example, if we believe that democracies are not only qualitatively different from autocracies, but also from anocracies, and we therefore introduce this new term into the literature, then it seems confusing to us that the exact same empirical property of a case (e.g. a Polity score of 7) qualifies a case as being a good instance of two conceptually distinct sets (e.g., democracy and anocracy). We are not sure how exactly G&M position themselves on these two issues. They do argue that partial membership in two or more mutually exclusive sets should be possible and they have no issue with such partial membership being higher than 0.5 in more than one set. We agree that partial membership in mutually exclusive sets is unproblematic—if and only if, however, a case’s membership does not exceed 0.5 in more than one of these sets. This is where we disagree with G&M. This allows two interpretations. G&M might claim that fuzzy sets do not establish any qualitative distinctions and that all is just a matter of degree. For reasons outlined above, this runs Qualitative & Multi-Method Research, Spring 2013