摘要
Abstract
Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content, and formalizing a notion of proof irrelevance. Indeed, semantically, the notion of a support is sometimes used as surrogate proposition asserting inhabitation of an indexed family. We give rules for bracket types in dependent type theory and provide complete semantics using regular categories. We show that dependent type theory with the unit type, strong extensional equality types, strong dependent sums, and bracket types is the internal type theory of regular categories, in the same way that the usual dependent type theory with dependent sums and products is the internal type theory of locally Cartesian closed categories. We also show how to interpret first-order logic in type theory with brackets, and we make use of the translation to compare type theory with logic. Specifically, we show that the propositions-as-types interpretation is complete with respect to a certain fragment of intuitionistic first-order logic, in the sense that a formula from the fragment is derivable in intuitionistic first-order logic if, and only if, its interpretation in dependent type theory is inhabited. As a consequence, a modified double-negation translation into type theory (without bracket types) is complete, in the same sense, for all of classical first-order logic.
摘要译文
抽象\r
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在常规类别图片因式分解正在回调稳定，所以他们在模型依赖型理论的自然模式运营。这种一元式构造[A]已经变成了以前在语法形式擦除的计算内容，并正式证明不相干的概念的一种方式。事实上，语义，支撑物的概念有时被用来作为一个索引的家庭替代命题的认定居住。我们给于依赖型支架理论类型的规则，并提供使用常规类别完整的语义。我们表明，依赖型理论与单元型，强拉伸平等类型，强依赖求和，并且托架的类型是常规类的内部类型的理论，在同样的方式，通常依赖型理论与依赖和与产品是内部键入本地笛卡儿闭范畴的理论。我们还表明如何解释一阶逻辑类型理论与支架，我们利用翻译到类型理论与逻辑相比较。具体地讲，我们表明，命题-作为类型解读完成相对于直觉一阶逻辑的某一片段，在从该片段的公式是衍生于直觉一阶逻辑，当且仅当该意义上说，其依赖型的理论解释是有人居住。因此，修改后的双重否定翻译成类型理论（无支架型）完成后，在同样的意义，对于所有经典的一阶逻辑的。
Steven Awodey1 ; ;rej Bauer2. Propositions as [Types][J]. Journal of Logic and Computation, 2004,14(4): 447-471