35,537 research outputs found
Theory Morphisms in Church's Type Theory with Quotation and Evaluation
is a version of Church's type theory with global
quotation and evaluation operators that is engineered to reason about the
interplay of syntax and semantics and to formalize syntax-based mathematical
algorithms. is a variant of that
admits undefined expressions, partial functions, and multiple base types of
individuals. It is better suited than as a logic for
building networks of theories connected by theory morphisms. This paper
presents the syntax and semantics of , defines a notion of
a theory morphism from one theory to another, and gives
two simple examples that illustrate the use of theory morphisms in .Comment: 17 page
Formalizing Mathematical Knowledge as a Biform Theory Graph: A Case Study
A biform theory is a combination of an axiomatic theory and an algorithmic
theory that supports the integration of reasoning and computation. These are
ideal for formalizing algorithms that manipulate mathematical expressions. A
theory graph is a network of theories connected by meaning-preserving theory
morphisms that map the formulas of one theory to the formulas of another
theory. Theory graphs are in turn well suited for formalizing mathematical
knowledge at the most convenient level of abstraction using the most convenient
vocabulary. We are interested in the problem of whether a body of mathematical
knowledge can be effectively formalized as a theory graph of biform theories.
As a test case, we look at the graph of theories encoding natural number
arithmetic. We used two different formalisms to do this, which we describe and
compare. The first is realized in , a version of Church's
type theory with quotation and evaluation, and the second is realized in Agda,
a dependently typed programming language.Comment: 43 pages; published without appendices in: H. Geuvers et al., eds,
Intelligent Computer Mathematics (CICM 2017), Lecture Notes in Computer
Science, Vol. 10383, pp. 9-24, Springer, 201
W.E. Vaughan, '<i>Murder Trials in Ireland, 1836-1914</i>'; Eleanor Gordon and Gwyneth Nair, '<i>Murder and Morality in Victorian Britain: the Story of Madeleine Smith</i>': Reviews
Penal modernism: an American tragedy
This paper discusses Whitman’s analysis of penal modernism. While I am in agreement with the central claim that penal modernism has been ignored and caricatured, I argue here that Whitman’s account of the penal modernist theory of judging must be understood in the context of a wider reframing of the social functions of the criminal law in penal modernism. This is explored by considering the unusual connection that the novel An American Tragedy (1925) has to the history of American criminal law, and the light that this can shed on the question of the meaning of penal modernism
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