Department of Psychology, Center for Cognitive Science, Rutgers University-New Brunswick, 152 Frelinghuysen Rd., Piscataway, NJ 08854, United States.
Cognition. 2012 Apr;123(1):61-83. doi: 10.1016/j.cognition.2011.12.008. Epub 2012 Jan 24.
Symbolic representation of environmental variables is a ubiquitous and often debated component of cognitive science. Yet notwithstanding centuries of philosophical discussion, the efficacy, scope, and validity of such representation has rarely been given direct consideration from a mathematical point of view. This paper introduces a quantitative measure of the effectiveness of symbolic representation, and develops formal constraints under which such representation is in fact warranted. The effectiveness of symbolic representation hinges on the probabilistic structure of the environment that is to be represented. For arbitrary probability distributions (i.e., environments), symbolic representation is generally not warranted. But in modal environments, defined here as those that consist of mixtures of component distributions that are narrow ("spiky") relative to their spreads, symbolic representation can be shown to represent the environment with a relatively negligible loss of information. Modal environments support propositional forms, logical relations, and other familiar features of symbolic representation. Hence the assumption that our environment is, in fact, modal is a key tacit assumption underlying the use of symbols in cognitive science.
符号表示法是认知科学中无处不在且经常被讨论的一个组成部分。然而，尽管经过了几个世纪的哲学讨论，从数学角度直接考虑这种表示法的功效、范围和有效性的情况却很少见。本文引入了一种衡量符号表示法有效性的定量方法，并提出了在何种形式约束下这种表示法实际上是有保证的。符号表示法的有效性取决于要表示的环境的概率结构。对于任意的概率分布（即环境），通常不保证符号表示法的合理性。但是，在模态环境中，可以证明符号表示法可以在相对较小的信息损失下表示环境，这里模态环境定义为由相对于其扩散程度较窄（“尖锐”）的组成分布的混合物组成的环境。模态环境支持命题形式、逻辑关系和符号表示法的其他常见特征。因此，假设我们的环境实际上是模态的，这是认知科学中使用符号的一个关键隐性假设。
