12 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
A Foundational View on Integration Problems
The integration of reasoning and computation services across system and
language boundaries is a challenging problem of computer science. In this
paper, we use integration for the scenario where we have two systems that we
integrate by moving problems and solutions between them. While this scenario is
often approached from an engineering perspective, we take a foundational view.
Based on the generic declarative language MMT, we develop a theoretical
framework for system integration using theories and partial theory morphisms.
Because MMT permits representations of the meta-logical foundations themselves,
this includes integration across logics. We discuss safe and unsafe integration
schemes and devise a general form of safe integration
Abstraction Logic: A New Foundation for (Computer) Mathematics
Abstraction logic is a new logic, serving as a foundation of mathematics. It
combines features of both predicate logic and higher-order logic: abstraction
logic can be viewed both as higher-order logic minus static types as well as
predicate logic plus operators and variable binding. We argue that abstraction
logic is the best foundational logic possible because it maximises both
simplicity and practical expressivity. This argument is supported by the
observation that abstraction logic has simpler terms and a simpler notion of
proof than all other general logics. At the same time, abstraction logic can
formalise both intuitionistic and classical abstraction logic, and is sound and
complete for these logics and all other logics extending deduction logic with
equality
Termos Singulares Indefinidos: Frege, Russell e a tradição matemática
É bem conhecida a divergência entre as posições de Gottlob Frege e Bertrand Russell com relação ao tratamento semântico dado a sentenças contendo termos singulares indefinidos, ou seja, termos singulares sem referência ou com referência ambÃgua, tais como ‘Papai Noel’ ou ‘o atual rei da França’ ou ‘1/0 ’ ou ‘√4’ ou ‘o autor de Principia Mathematica’. Para Frege, as sentenças da linguagem natural que contêm termos indefinidos não formam declarações e portanto não são nem verdadeiras nem falsas. Já para as sentenças da matemática, Frege defende que elas precisam ser corrigidas através da convenção forçada de uma referência não ambÃgua. Russell, por outro lado, aceita os termos indefinidos e propõe, através de sua teoria das descrições definidas, uma maneira de avaliar as sentenças em que eles ocorrem; e Quine amplia a teoria de Russell para abranger também os nomes com problemas de referência. Na prática da matemática são comuns os termos singulares indefinidos, sem referência, tais como ‘1/0 ’, ou com referência ambÃgua, tais como ‘√4’. Apesar de não haver uma sistematização rigorosa desta situação entre os matemáticos, há, no entanto, um conjunto de regras convencionais que tradicionalmente costumam ser aplicadas no tratamento matemático dos termos indefinidos. Nossa proposta é tomar a convenção matemática como inspiração e modelo para apresentar uma interpretação semântica formal para as descrições definidas e os nomes e utilizá-la como um argumento que favorece a abordagem de Russell relativamente à de Frege
Free Higher-Order Logic - Notion, Definition and Embedding in HOL
Free logics are a family of logics that are free of any existential assumptions. This family can roughly be divided into positive, negative, neutral and supervaluational free logic whose semantics differ in the way how nondenoting terms are treated. While there has been remarkable work done concerning the definition of free first-order logic, free higher-order logic has not been addressed thoroughly so far. The purpose of this thesis is, firstly, to give a notion and definition of free higher-order logic based on simple type theory and, secondly, to propose faithful shallow semantical embeddings of free higher-order logic into classical higher order logic found on this definition. Such embeddings can then effectively be utilized to enable the application of powerful state-of-the-art higher-order interactive and automated theorem provers for the formalization and verification and also the further development of increasingly important free logical theories