16,170 research outputs found
Towards an Intelligent Tutor for Mathematical Proofs
Computer-supported learning is an increasingly important form of study since
it allows for independent learning and individualized instruction. In this
paper, we discuss a novel approach to developing an intelligent tutoring system
for teaching textbook-style mathematical proofs. We characterize the
particularities of the domain and discuss common ITS design models. Our
approach is motivated by phenomena found in a corpus of tutorial dialogs that
were collected in a Wizard-of-Oz experiment. We show how an intelligent tutor
for textbook-style mathematical proofs can be built on top of an adapted
assertion-level proof assistant by reusing representations and proof search
strategies originally developed for automated and interactive theorem proving.
The resulting prototype was successfully evaluated on a corpus of tutorial
dialogs and yields good results.Comment: In Proceedings THedu'11, arXiv:1202.453
Mill on logic
Working within the broad lines of general consensus that mark out the core features of John Stuart Mill’s (1806–1873) logic, as set forth in his A System of Logic (1843–1872), this chapter provides an introduction to Mill’s logical theory by reviewing his position on the relationship between induction and deduction, and the role of general premises and principles in reasoning. Locating induction, understood as a kind of analogical reasoning from particulars to particulars, as the basic form of inference that is both free-standing and the sole load-bearing structure in Mill’s logic, the foundations of Mill’s logical system are briefly inspected. Several naturalistic features are identified, including its subject matter, human reasoning, its empiricism, which requires that only particular, experiential claims can function as basic reasons, and its ultimate foundations in ‘spontaneous’ inference. The chapter concludes by comparing Mill’s naturalized logic to Russell’s (1907) regressive method for identifying the premises of mathematics
Some blunt instruments of dogmatic logic: Sextus Empiricus’s sceptical attack
Within a sort of conceptually homogeneous logical-epistemological arsenal that reflects a perspective marked by the dichotomy true/false, I would like to focus on one of the ‘logical’ sections of Sextus Empiricus's Outlines of Pyrrhonism, book II, namely: division, whole/parts, genus/species, accidents (PH II 213-228). These are unique in Sextus’ work and hence find no parallels in the more meticulous analysis provided in M VII and VIII. This uniqueness does not merely concern the dogmatic theories pertaining to the notions of syllogism, induction/definition and sophism, but extends to a range of points made with regard to logical-demonstrative argumentation, and which appear to play a leading role in the doctrines that Sextus seeks to refute
The First-Order Hypothetical Logic of Proofs
The Propositional Logic of Proofs (LP) is a modal logic in which the modality □A is revisited as [[t]]A , t being an expression that bears witness to the validity of A . It enjoys arithmetical soundness and completeness, can realize all S4 theorems and is capable of reflecting its own proofs ( ⊢A implies ⊢[[t]]A , for some t ). A presentation of first-order LP has recently been proposed, FOLP, which enjoys arithmetical soundness and has an exact provability semantics. A key notion in this presentation is how free variables are dealt with in a formula of the form [[t]]A(i) . We revisit this notion in the setting of a Natural Deduction presentation and propose a Curry–Howard correspondence for FOLP. A term assignment is provided and a proof of strong normalization is given.Fil: Steren, Gabriela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Bonelli, Eduardo Augusto. Universidad Nacional de Quilmes. Departamento de Ciencia y Tecnología; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin
A Primer on the Tools and Concepts of Computable Economics
Computability theory came into being as a result of Hilbert's attempts to meet Brouwer's challenges, from an intuitionistc and constructive standpoint, to formalism as a foundation for mathematical practice. Viewed this way, constructive mathematics should be one vision of computability theory. However, there are fundamental differences between computability theory and constructive mathematics: the Church-Turing thesis is a disciplining criterion in the former and not in the latter; and classical logic - particularly, the law of the excluded middle - is not accepted in the latter but freely invoked in the former, especially in proving universal negative propositions. In Computable Economic an eclectic approach is adopted where the main criterion is numerical content for economic entities. In this sense both the computable and the constructive traditions are freely and indiscriminately invoked and utilised in the formalization of economic entities. Some of the mathematical methods and concepts of computable economics are surveyed in a pedagogical mode. The context is that of a digital economy embedded in an information society
Some observations on the logical foundations of inductive theorem proving
In this paper we study the logical foundations of automated inductive theorem
proving. To that aim we first develop a theoretical model that is centered
around the difficulty of finding induction axioms which are sufficient for
proving a goal.
Based on this model, we then analyze the following aspects: the choice of a
proof shape, the choice of an induction rule and the language of the induction
formula. In particular, using model-theoretic techniques, we clarify the
relationship between notions of inductiveness that have been considered in the
literature on automated inductive theorem proving. This is a corrected version
of the paper arXiv:1704.01930v5 published originally on Nov.~16, 2017
Dialectical Strategic Planning in Aristotle
The purpose of this paper is to give an account and a rational reconstruction of the heuristic advice provided by Aristotle in the Topics and Prior Analytics in regard to the difficulty or ease of strategic planning in the context of a dialectical dialogue. The general idea is that a Questioner can foresee what his refutational syllogism would have to look like given the character of the thesis defended by the Answerer, and therefore plan accordingly. A rational reconstruction of this advice will be attempted from three perspectives: strategic planning based on the acceptability of Answerer’s thesis, strategic planning based on the predicational form of the thesis, strategic planning based on the logical form of the thesis. In addition, we will provide an illustration of the potential of this heuristic advice as we apply it to the interpretation of a fragment from Plato, presuming that, in a similar way, a reading of this kind might be more generally applicable in the interpretation of
the Platonic dialogues
Logic in the Tractatus
I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named.
There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably infinite, then the property of being a tautology is \Pi^1_1-complete. But third, it is only granted the assumption of countability that the class of tautologies is \Sigma_1-definable in set theory.
Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects
Synthetic biology—putting engineering into biology
Synthetic biology is interpreted as the engineering-driven building of increasingly complex biological entities for novel applications. Encouraged by progress in the design of artificial gene networks, de novo DNA synthesis and protein engineering, we review the case for this emerging discipline. Key aspects of an engineering approach are purpose-orientation, deep insight into the underlying scientific principles, a hierarchy of abstraction including suitable interfaces between and within the levels of the hierarchy, standardization and the separation of design and fabrication. Synthetic biology investigates possibilities to implement these requirements into the process of engineering biological systems. This is illustrated on the DNA level by the implementation of engineering-inspired artificial operations such as toggle switching, oscillating or production of spatial patterns. On the protein level, the functionally self-contained domain structure of a number of proteins suggests possibilities for essentially Lego-like recombination which can be exploited for reprogramming DNA binding domain specificities or signaling pathways. Alternatively, computational design emerges to rationally reprogram enzyme function. Finally, the increasing facility of de novo DNA synthesis—synthetic biology’s system fabrication process—supplies the possibility to implement novel designs for ever more complex systems. Some of these elements have merged to realize the first tangible synthetic biology applications in the area of manufacturing of pharmaceutical compounds.
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