125,230 research outputs found
Axiomatizations of arithmetic and the first-order/second-order divide
It is often remarked that first-order Peano Arithmetic is non-categorical but deductively well-behaved, while second-order Peano Arithmetic is categorical but deductively ill-behaved. This suggests that, when it comes to axiomatizations of mathematical theories, expressive power and deductive power may be orthogonal, mutually exclusive desiderata. In this paper, I turn to Hintikkaâs (Philos Top 17(2):69â90, 1989) distinction between descriptive and deductive approaches in the foundations of mathematics to discuss the implications of this observation for the first-order logic versus second-order logic divide. The descriptive approach is illustrated by Dedekindâs âdiscoveryâ of the need for second-order concepts to ensure categoricity in his axiomatization of arithmetic; the deductive approach is illustrated by Fregeâs Begriffsschrift project. I argue that, rather than suggesting that any use of logic in the foundations of mathematics is doomed to failure given the impossibility of combining the descriptive approach with the deductive approach, what this apparent predicament in fact indicates is that the first-order versus second-order divide may be too crude to investigate what an adequate axiomatization of arithmetic should look like. I also conclude that, insofar as there are different, equally legitimate projects one may engage in when working on the foundations of mathematics, there is no such thing as the One True Logic for this purpose; different logical systems may be adequate for different projects
Classical Predicative Logic-Enriched Type Theories
A logic-enriched type theory (LTT) is a type theory extended with a primitive
mechanism for forming and proving propositions. We construct two LTTs, named
LTTO and LTTO*, which we claim correspond closely to the classical predicative
systems of second order arithmetic ACAO and ACA. We justify this claim by
translating each second-order system into the corresponding LTT, and proving
that these translations are conservative. This is part of an ongoing research
project to investigate how LTTs may be used to formalise different approaches
to the foundations of mathematics.
The two LTTs we construct are subsystems of the logic-enriched type theory
LTTW, which is intended to formalise the classical predicative foundation
presented by Herman Weyl in his monograph Das Kontinuum. The system ACAO has
also been claimed to correspond to Weyl's foundation. By casting ACAO and ACA
as LTTs, we are able to compare them with LTTW. It is a consequence of the work
in this paper that LTTW is strictly stronger than ACAO.
The conservativity proof makes use of a novel technique for proving one LTT
conservative over another, involving defining an interpretation of the stronger
system out of the expressions of the weaker. This technique should be
applicable in a wide variety of different cases outside the present work.Comment: 49 pages. Accepted for publication in special edition of Annals of
Pure and Applied Logic on Computation in Classical Logic. v2: Minor mistakes
correcte
An institution for Alloy and its translation to second-order logic
Publicado em "Integration of reusable systems". ISBN 978-3-319-04716-4Lightweight formal methods, of which Alloy is a prime example, combine the rigour of mathematics without compromising simplicity of use and suitable tool support. In some cases, however, the verification of safety or mission critical software entails the need for more sophisticated technologies, typically based on theorem provers. This explains a number of attempts to connect Alloy to specific theorem provers documented in the literature. This chapter, however, takes a different perspective: instead of focusing on one more combination of Alloy with still another prover, it lays out the foundations to fully integrate this system in the Hets platform which supports a huge network of logics, logic translators and provers. This makes possible for Alloy specifications to âborrowâ the power of several, non dedicated proof systems. The chapter extends the authorsâ previous work on this subject by developing in full detail the semantical foundations for this integration, including a formalisation of Alloy as an institution, and introducing a new, more general translation of the latter to second-order logic.(undefined
Descriptive Complexity, Computational Tractability, and the Logical and Cognitive Foundations of Mathematics
In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational models of human cognitive capacities. One interesting area of computational complexity theory is descriptive complexity, which connects the expressive strength of systems of logic with the computational complexity classes. In descriptive complexity theory, it is established that only first-order (classical) systems are connected to P, or one of its subclasses. Consequently, second-order systems of logic are considered to be computationally intractable, and may therefore seem to be unfit to model human cognitive capacities. This would be problematic when we think of the role of logic as the foundations of mathematics. In order to express many important mathematical concepts and systematically prove theorems involving them, we need to have a system of logic stronger than classical first-order logic. But if such a system is considered to be intractable, it means that the logical foundation of mathematics can be prohibitively complex for human cognition. In this paper I will argue, however, that this problem is the result of an unjustified direct use of computational complexity classes in cognitive modelling. Placing my account in the recent literature on the topic, I argue that the problem can be solved by considering computational complexity for humanly relevant problem solving algorithms and input sizes.Peer reviewe
Independence in Model Theory and Team Semantics
The subject of this doctoral thesis is the mathematical theory of independence, and its various manifestations in logic and mathematics. The topics covered in this doctoral thesis range from model theory and combinatorial geometry, to database theory, quantum logic and probability logic. This study has two intertwined centres: - classification theory, independence calculi and combinatorial geometry (papers I-IV); - new perspectives in team semantics (papers V-VII).
The first topic is a classical topic in model theory, which we approach from different directions (implication problems, abstract elementary classes, unstable first-order theories). The second topic is a relatively new logical framework where to study non-classical logical phenomena (dependence and independence, uncertainty, probabilistic reasoning, quantum foundations). Although these two centres seem to be far apart, we will see that they are linked to each others in various ways, under the guiding thread of independence
Kriesel and Wittgenstein
Georg Kreisel (15 September 1923 - 1 March 2015) was a formidable mathematical
logician during a formative period when the subject was becoming
a sophisticated field at the crossing of mathematics and logic. Both with his
technical sophistication for his time and his dialectical engagement with mandates,
aspirations and goals, he inspired wide-ranging investigation in the metamathematics
of constructivity, proof theory and generalized recursion theory.
Kreisel's mathematics and interactions with colleagues and students have been
memorably described in Kreiseliana ([Odifreddi, 1996]). At a different level of
interpersonal conceptual interaction, Kreisel during his life time had extended
engagement with two celebrated logicians, the mathematical Kurt GĂśdel and
the philosophical Ludwig Wittgenstein. About GĂśdel, with modern mathematical
logic palpably emanating from his work, Kreisel has reflected and written
over a wide mathematical landscape. About Wittgenstein on the other hand,
with an early personal connection established Kreisel would return as if with
an anxiety of influence to their ways of thinking about logic and mathematics,
ever in a sort of dialectic interplay. In what follows we draw this out through
his published essaysâand one letterâboth to elicit aspects of influence in his
own terms and to set out a picture of Kreisel's evolving thinking about logic
and mathematics in comparative relief.Accepted manuscrip
Hilbert's Program Then and Now
Hilbert's program was an ambitious and wide-ranging project in the philosophy
and foundations of mathematics. In order to "dispose of the foundational
questions in mathematics once and for all, "Hilbert proposed a two-pronged
approach in 1921: first, classical mathematics should be formalized in
axiomatic systems; second, using only restricted, "finitary" means, one should
give proofs of the consistency of these axiomatic systems. Although Godel's
incompleteness theorems show that the program as originally conceived cannot be
carried out, it had many partial successes, and generated important advances in
logical theory and meta-theory, both at the time and since. The article
discusses the historical background and development of Hilbert's program, its
philosophical underpinnings and consequences, and its subsequent development
and influences since the 1930s.Comment: 43 page
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